Multiscale Computations for Flow and Transport in Porous Mediausers.cms.caltech.edu/~hou/papers/Lecture_Hou.pdf · and multiresolution. The lectures can be roughly divided into six

Multiscale Computations for Flow and Transport in Porous Media

Thomas Y. Hou

Applied Mathematics, 217-50, Caltech, Pasadena, CA 91125, USA. Email: [email protected].

Abstract. Many problems of fundamental and practical importance have multiple scale solutions. The direct numer-ical solution of multiple scale problems is difficult to obtain even with modern supercomputers. The major difficultyof direct solutions is the scale of computation. The ratio between the largest scale and the smallest scale could be aslarge as 105 in each space dimension. From an engineering perspective, it is often sufficient to predict the macroscopicproperties of the multiple-scale systems, such as the effective conductivity, elastic moduli, permeability, and eddydiffusivity. Therefore, it is desirable to develop a method that captures the small scale effect on the large scales, butdoes not require resolving all the small scale features. This paper reviews some of the recent advances in developingsystematic multiscale methods with particular emphasis on multiscale finite element methods with applications toflow and transport in heterogeneous porous media. This manuscript is not intended to be a general survey paper onthis topic. The discussion is limited by the scope of the lectures and expertise of the author.

1 Introduction

Many problems of fundamental and practical importance have multiple scale solutions. Composite materials,porous media, and turbulent transport in high Reynolds number flows are examples of this type. A com-plete analysis of these problems is extremely difficult. For example, the difficulty in analyzing groundwatertransport is mainly caused by the heterogeneity of subsurface formations spanning over many scales. Thisheterogeneity is often represented by the multiscale fluctuations in the permeability of media. For compositematerials, the dispersed phases (particles or fibers), which may be randomly distributed in the matrix, giverise to fluctuations in the thermal or electrical conductivity; moreover, the conductivity is usually discontin-uous across the phase boundaries. In turbulent transport problems, the convective velocity field fluctuatesrandomly and contains many scales depending on the Reynolds number of the flow.

The direct numerical solution of multiple scale problems is difficult even with the advent of supercom-puters. The major difficulty of direct solutions is the scale of computation. For groundwater simulations, itis common that millions of grid blocks are involved, with each block having a dimension of tens of meters,whereas the permeability measured from cores is at a scale of several centimeters. This gives more than 105

degrees of freedom per spatial dimension in the computation. Therefore, a tremendous amount of computermemory and CPU time are required, and this can easily exceed the limit of today’s computing resources. Thesituation can be relieved to some degree by parallel computing; however, the size of the discrete problem isnot reduced. The load is merely shared by more processors with more memory. Whenever one can afford toresolve all the small scale features of a physical problem, direct solutions provide quantitative information ofthe physical processes at all scales. On the other hand, from an engineering perspective, it is often sufficientto predict the macroscopic properties of the multiscale systems, such as the effective conductivity, elasticmoduli, permeability, and eddy diffusivity. Therefore, it is desirable to develop a method that captures thesmall scale effect on the large scales, but does not require resolving all the small scale features. Upscalingprocedures have been commonly applied for this purpose and are effective in many cases. More recently, anumber of multiscale techniques have been developed and successfully applied to various areas, e.g., porousmedia flows. The main idea of upscaling techniques is to form coarse-scale equations with a prescribed an-alytical form that may differ from the underlying fine-scale equations. In multiscale methods, the fine-scaleinformation is carried throughout the simulation and the coarse-scale equations are generally not expressedanalytically, but rather formed and solved numerically.

The purpose of this lecture note is to review some recent advances in developing multiscale finite ele-ment (volume) methods for flow and transport in strongly heterogeneous porous media. Extra effort is madein developing a multiscale computational method that can be potentially used for practical multiscale forproblems with a large range of nonseparable scales. Substantial progress has been made in recent years by

2 Thomas Y. Hou

combining modern mathematical techniques such as multiscale analysis, random sampling, and adaptivityand multiresolution. The lectures can be roughly divided into six parts. In Section 2, we will review somehomogenization theory for elliptic and hyperbolic equations as well as for incompressible flows. This homog-enization theory provides the critical guideline for designing effective multiscale methods. In Section 3, wediscuss numerical homogenization based on sampling techniques. In Section 4, we discuss some recent devel-opments of multiscale finite element methods. We also discuss the issue of upscaling one-phase, two-phaseflows through heterogeneous porous media and the use of limited global information in multiscale finiteelement methods. In Section 5, we discuss the generalization of the multiscale finite element methods to non-linear partial differential equations. In Section 6, we will consider multiscale simulations of two-phase flowimmiscible flows using a flow-based adaptive coordinate system. There are many other multiscale methodswhich we will not cover due to the limited scope of these lectures. The above methods are chosen becausethey are similar philosophically and the materials complement each other very well. This paper is not in-tended to be a detailed survey of all available multiscale methods. The discussion is limited by scope of thelectures and expertise of the author.

2 Review of Homogenization Theory

In this section, we will review some classical homogenization theory for elliptic and hyperbolic PDEs. Thishomogenization theory will play an essential role in designing effective multiscale numerical methods forpartial differential equations with multiscale solutions.

2.1 Homogenization Theory for Elliptic Problems

Consider the second order elliptic equation

L(uε) ≡ − ∂

∂xi

(

aij (x/ε)∂

∂xj

)

uε + a0(x/ε)uε = f, uε|∂Ω = 0, (2.1)

where aij(y) and a0(y) are 1-periodic in both variables of y, and satisfy aij(y)ξiξj ≥ αξiξi, with α > 0, anda0 > α0 > 0. Here we have used the Einstein summation notation, i.e. repeated index means summationwith respect to that index.

This model equation represents a common difficulty shared by several physical problems. For porousmedia, it is the pressure equation through Darcy’s law, the coefficient aε representing the permeabilitytensor. For composite materials, it is the steady heat conduction equation and the coefficient aε representsthe thermal conductivity. For steady transport problems, it is a symmetrized form of the governing equation.In this case, the coefficient aε is a combination of transport velocity and viscosity tensor.

Homogenization theory is to study the limiting behavior uε → u as ε → 0. The main task is to find thehomogenized coefficients, a∗ij and a∗0, and the homogenized equation for the limiting solution u

− ∂

∂xi

(

a∗ij∂

∂xj

)

u+ a∗0u = f, u|∂Ω = 0. (2.2)

Define the L2 and H1 norms over Ω as follows

‖v‖20 =

∫

Ω

|v|2 dx, ‖v‖21 = ‖v‖2

0 + ‖∇v‖20. (2.3)

Further, we define the bilinear form

aε(u, v) =

∫

Ω

aεi,j(x)∂u

∂xj

∂v

∂xidx+

∫

Ω

aε0uv dx. (2.4)

It is easy to show thatc1‖u‖2

1 ≤ aε(u, u) ≤ c2‖u‖21, (2.5)

Multiscale Computations for Flow and Transport in Porous Media 3

with c1 = min(α, α0), c2 = max(‖aij‖∞, ‖a0‖∞).The elliptic problem can also be formulated as a variational problem: find uε ∈ H1

0

aε(uε, v) = (f, v), for all v ∈ H10 (Ω), (2.6)

where (f, v) is the usual L2 inner product,∫

Ω fv dx.

Special Case: One-Dimensional Problem Let Ω = (x0, x1) and take a0 = 0. We have

− d

dx

(

a(x/ε)duεdx

)

= f, in Ω , (2.7)

where uε(x0) = uε(x1) = 0, and a(y) > α0 > 0 is y-periodic with period y0.By taking v = uε in the bilinear form, we have

‖uε‖1 ≤ c.

Therefore one can extract a subsequence, still denoted by uε, such that

uε u in H10 (Ω) weakly. (2.8)

On the other hand, we notice that

aε m(a) =1

y0

∫ y0

0

a(y) dy in L∞(Ω) weak star. (2.9)

It is tempting to conclude that u satisfies:

− d

dx

(

m(a)du

dx

)

= f,

where m(a) = 1y0

∫ y00 a(y) dy is the arithmetic mean of a. However, this is not true. To derive the correct

answer, we introduce

ξε = aεduε

dx.

Since aε is bounded, and uεx is bounded in L2(Ω), so ξε is bounded in L2(Ω). Moreover, since − dξε

dx = f , wehave ξε ∈ H1(Ω). Thus we get

ξε → ξ in L2(Ω) strongly,

so that1

aεξε → m(1/a)ξ in L2(Ω) weakly.

Further, we note that 1aε ξ

ε = duε

dx . Therefore, we arrive at

du

dx= m(1/a)ξ.

On the other hand, − dξε

dx = f implies − dξdx = f . This gives

− d

dx

(

1

m(1/a)

du

dx

)

= f. (2.10)

This is the correct homogenized equation for u. Note that a∗ = 1m(1/a) is the harmonic average of aε. It is in

general not equal to the arithmetic average aε = m(a).

4 Thomas Y. Hou

Multiscale Asymptotic Expansions. The above analysis does not generalize to multi-dimensions. Inthis subsection, we introduce the multiscale expansion technique in deriving homogenized equations. Thistechnique is very effective and can be used in a number of applications.

We shall look for uε(x) in the form of asymptotic expansion

uε(x) = u0(x, x/ε) + εu1(x, x/ε) + ε2u2(x, x/ε) + · · · , (2.11)

where the functions uj(x, y) are double periodic in y with period 1.Denote by Aε the second order elliptic operator

Aε = − ∂

∂xi

(

aij (x/ε)∂

∂xj

)

. (2.12)

When differentiating a function φ(x, x/ε) with respect to x, we have

∂

∂xj=

∂

∂xj+

1

ε

∂

∂yj,

where y is evaluated at y = x/ε. With this notation, we can expand Aε as follows

Aε = ε−2A1 + ε−1A2 + ε0A3, (2.13)

where

A1 = − ∂

∂yi

(

aij(y)∂

∂yj

)

, (2.14)

A2 = − ∂

∂yi

(

aij(y)∂

∂xj

)

− ∂

∂xi

(

aij(y)∂

∂yj

)

, (2.15)

A3 = − ∂

∂xi

(

aij(y)∂

∂xj

)

+ a0 . (2.16)

Substituting the expansions for uε and Aε into Aεuε = f , and equating the terms of the same power, we get

A1u0 = 0, (2.17)

A1u1 + A2u0 = 0, (2.18)

A1u2 + A2u1 +A3u0 = f. (2.19)

Equation (2.17) can be written as

− ∂

∂yi

(

aij(y)∂

∂yj

)

u0(x, y) = 0, (2.20)

where u0 is periodic in y. The theory of second order elliptic PDEs [57] implies that u0(x, y) is independentof y, i.e. u0(x, y) = u0(x). This simplifies equation (2.18) for u1,

− ∂

∂yi

(

aij(y)∂

∂yj

)

u1 =

(

∂

∂yiaij(y)

)

∂u

∂xj(x).

Define χj = χj(y) as the solution to the following cell problem

∂

∂yi

(

aij(y)∂

∂yj

)

χj =∂

∂yiaij(y) , (2.21)

where χj is double periodic in y. The general solution of equation (2.18) for u1 is then given by

u1(x, y) = −χj(y) ∂u∂xj

(x) + u1(x) . (2.22)


Finally, we note that the equation for u2 is given by

∂

∂yi

(

aij(y)∂

∂yj

)

u2 = A2u1 +A3u0 − f . (2.23)

The solvability condition implies that the right hand side of (2.23) must have mean zero in y over one periodiccell Y = [0, 1]× [0, 1], i.e.

∫

Y

(A2u1 +A3u0 − f) dy = 0.

This solvability condition for second order elliptic PDEs with periodic boundary condition [57] requires thatthe right hand side of equation (2.23) have mean zero with respect to the fast variable y. This solvabilitycondition gives rise to the homogenized equation for u:

− ∂

∂xi

(

a∗ij∂

∂xj

)

u+m(a0)u = f , (2.24)

where m(a0) = 1|Y |∫

Ya0(y) dy and

a∗ij =1

|Y |

(∫

Y

(aij − aik∂χj

∂yk) dy

)

. (2.25)

Justification of formal expansions The above multiscale expansion is based on a formal asymptoticanalysis. However, we can justify its convergence rigorously.

Let zε = uε − (u+ εu1 + ε2u2). Applying Aε to zε, we get

Aεzε = −εrε ,

where rε = A2u2 +A3u1 + εA3u2. If f is smooth enough, so is u2. Thus we have ‖rε‖∞ ≤ c.On the other hand, we have

zε|∂Ω = −(εu1 + ε2u2)|∂Ω.

Thus, we obtain‖zε‖L∞(∂Ω) ≤ cε.

It follows from the maximum principle [57] that

‖zε‖L∞(Ω) ≤ cε

and therefore we conclude that‖uε − u‖L∞(Ω) ≤ cε.

Boundary Corrections The above asymptotic expansion does not take into account the boundary condi-tion of the original elliptic PDEs. If we add a boundary correction, we can obtain higher order approximations.

Let θε ∈ H1(Ω) denote the solution to

∇x · aε∇xθε = 0 in Ω, θε = u1(x, x/ε) on ∂Ω.

Then we have(uε − (u+ εu1(x, x/ε) − εθε)) |∂Ω = 0.

Moskow and Vogelius [83] have shown that

‖uε − u− εu1(x, x/ε) + εθε‖0 ≤Cωε1+ω‖u‖2+ω,

‖uε − u− εu1(x, x/ε) + εθε‖1 ≤Cε‖u‖2,(2.26)

where we assume u ∈ H2+ω(Ω) with 0 ≤ ω ≤ 1, and Ω is assumed to be a bounded, convex curvilinearpolygon of class C∞. This improved estimate will be used in the convergence analysis of the multiscale finiteelement method to be presented in Section 4.

6 Thomas Y. Hou

2.2 Homogenization for hyperbolic problems

In this subsection, we will review some homogenization theory for semilinear hyperbolic systems. As wewill see below, homogenization for hyperbolic problems is very different from that for elliptic problems. Thephenomena are also very rich.

Consider the semilinear Carleman equations [23]:

∂uε∂t

+∂uε∂x

= v2ε − u2

ε,

∂vε∂t

− ∂vε∂x

= u2ε − v2

ε ,

with oscillatory initial data, uε(x, 0) = uε0(x), vε(x, 0) = vε0(x).Assume that the initial conditions are positive and bounded. Then it can be shown that there exists a

unique bounded solution for all times. Thus we can extract a subsequence of uε and vε such that uε u andvε v as ε→ 0.

Denote um as the weak limit of umε , and vm as the weak limit of vmε . By taking the weak limit of bothsides of the equations, we get

∂u1

∂t+∂u1

∂x= v2 − u2,

∂v1∂t

− ∂v1∂x

= u2 − v2.

By multiplying the Carleman equations by uε and vε respectively, we get

∂u2ε

∂t+∂u2

ε

∂x= 2uεv

2ε − 2u3

ε,

∂v2ε

∂t+∂v2

ε

∂x= 2vεu

2ε − 2v3

ε .

Thus the weak limit of u2ε depends on the weak limit of u3

ε and the weak limit of uεv2ε .

Denote by wε as the weak limit of wε. To obtain a closure, we would like to express uεv2ε in terms of the

product uε and v2ε . This is not possible in general. In this particular case, we can use the Div-Curl Lemma

[84,85,97] to obtain a closure.The Div-Curl Lemma. Let Ω be an open set of R

N and uε and vε be two sequences such that

uε u, in(

L2(Ω))N

weakly,

vε v, in(

L2(Ω))N

weakly.

Further, we assume that

div uε is bounded in L2(Ω)( or compact in H−1(Ω)) ,

curl vε is bounded in(

L2(Ω))N2

(or compact in(

H−1(Ω))N2

).

Let 〈·, ·〉 denote the inner product in RN , i.e.

〈u,v〉 =

N∑

i=1

uivi.

Then we have〈uε · vε〉〈u · v〉 weakly. (2.27)

Remark 2.1. We remark that the Div-Curl Lemma is the simplest form of the more general CompensatedCompactness Theory developed by Tartar [97] and Murat [84,85].


Applying the Div-Curl Lemma to (uε, uε) and (v2ε , v

2ε) in the space-time domain, one can show that

uεv2ε = uε v2

ε . Similarly, one can show that u2εvε = u2

ε vε. Using this fact, Tartar [98] obtained the followinginfinite hyperbolic system for um and vm [98]:

∂um∂t

+∂um∂x

= mum−1v2 −mum+1,

∂vm∂t

− ∂vm∂x

= mvm−1u2 −mvm+1.

Note that the weak limit of umε , um, depends on the weak limit of um+1ε , um+1. Similarly, vm depends

on vm+1. Thus one cannot obtain a closed system for the weak limits uε and vε by a finite system. This is ageneric phenomenon for nonlinear partial differential equations with microstructure. It is often referred to asthe closure problem. On the other hand, for the Carleman equations, Tartar showed that the infinite systemis hyperbolic and the system is well-posed.

The situation is very different for a 3 × 3 system of Broadwell type [22]:

∂uε∂t

+∂uε∂x

= w2ε − uεvε, (2.28)

∂vε∂t

− ∂vε∂x

= w2ε − uεvε, (2.29)

∂wε∂t

+ α∂wε∂x

= uεvε − w2ε , (2.30)

with oscillatory initial data, uε(x, 0) = uε0(x), vε(x, 0) = vε0(x) and wε(x, 0) = wε0(x). When α = 0, theabove system reduces to the original Broadwell model. We will refer to the above system as the generalizedBroadwell model.

Note that in the generalized Broadwell model, the right hand side of the w-equation depends on theproduct of uv. If we try to obtain an evolution equation for w2

ε , it will depend on the triple product uεvεwε.The Div-Curl Lemma cannot be used here to characterize the weak limit of this triple product in terms ofthe weak limits of uε, vε and wε.

Assume the initial oscillations are periodic, i.e.

uε0 = u0(x, x/ε), vε0 = v0(x, x/ε), w

ε0 = w0(x, x/ε).

where u0(x, y), v0(x, y), w0(x, y) are 1-periodic in y.

There are two cases to consider.

Case 1. α = m/n is a rational number. Let U(x, y, t), V (x, y, t),W (x, y, t) be the homogenized solutionwhich satisfies

Ut + Ux =∫ 1

0 W2 dy − U

∫ 1

0 V dy,

Vt − Vx =∫ 1

0W 2 dy − U

∫ 1

0V dy,

Wt + αWx = −W 2 +1

n

∫ n

0

U(x, y + (α− 1)z, t)V (x, y + (α+ 1)z, t) dz,

where U |t=0 = u0(x, y), V |t=0 = v0(x, y) and W |t=0 = w0(x, y). Then we have

‖uε(x, t) − U(x, x−tε , t)‖L∞ ≤ Cε,

‖vε(x, t) − V (x, x+tε , t)‖L∞ ≤ Cε,

‖wε(x, t) −W (x, x−αtε , t)‖L∞ ≤ Cε.

8 Thomas Y. Hou

Case 2. α is an irrational number. Let U(x, y, t), V (x, y, t),W (x, y, t) be the homogenized solution whichsatisfies

Ut + Ux =∫ 1

0W 2 dy − U

∫ 1

0V dy,

Vt − Vx =∫ 1

0 W2 dy − U

∫ 1

0 V dy,

Wt + αWx = −W 2 +(

∫ 1

0 U dy)(

∫ 1

0 V dy)

.

where U |t=0 = u0(x, y), V |t=0 = v0(x, y) and W |t=0 = w0(x, y). Then we have

‖uε(x, t) − U(x, x−tε , t)‖L∞ ≤ Cε,

‖vε(x, t) − V (x, x+tε , t)‖L∞ ≤ Cε,

‖wε(x, t) −W (x, x−αtε , t)‖L∞ ≤ Cε.

We refer the reader to [59] for the proof of the above results.Note that when α is a rational number, the interaction of uε and vε can generate a high frequency

contribution to wε. This is not the case when α is an irrational number. The rational α case corresponds toa resonance interaction.

The derivation and analysis of the above results rely on the following two Lemmas:

Lemma 2.1. Let f(x), g(x, y) ∈ C1. Assume that g(x, y) is n-periodic in y, then we have∫ b

a

f(x)g(x, x/ε) dx =

∫ b

a

f(x)

(

1

n

∫ n

0

g(x, y) dy

)

dx+O(ε).

Lemma 2.2. Let f(x, y, z) ∈ C1. Assume that f(x, y, z) is 1-periodic in y and z. If γ2/γ1 is an irrationalnumber, then we have

∫ b

a

f(

x, x1+γ1xε , x2+γ1x

ε

)

dx =

∫ b

a

(∫ 1

0

∫ 1

0

f(x, y, z) dy dz

)

dx+O(ε).

The proof uses some basic ergodic theory. It can be seen easily by expanding in Fourier series in theperiodic variables [59]. For the sake of completeness, we present a simple proof of the above homogenizationresult for the case of α = 0 in the next subsection.

Homogenization of the Broadwell Model In this subsection, we give a simple proof of the homogeniza-tion result in the special case of α = 0. The homogenized equations can be derived by multiscale asymptoticexpansions [81].

Consider the Broadwell model

∂tu+ ∂xu = w2 − uv in R × (0, T ), (2.31)

∂tv − ∂xv = w2 − uv in R × (0, T ), (2.32)

∂tw = uv − w2 in R × (0, T ), (2.33)

with oscillatory initial values

u(x, 0) = u0(x,xε ), v(x, 0) = v0(x,

xε ), w(x, 0) = w0(x,

xε ), (2.34)

where u0(x, y), v0(x, y), w0(x, y) are 1-periodic in y. We introduce an extra variable, y, to describe the fastvariable, x/ε. Let the solution of the homogenized equation be U(x, y, t), V (x, y, t),W (x, y, t) which satisfies

∂tU + ∂xU + U

∫ 1

0

V dy −∫ 1

0

W 2 dy = 0 in R × (0, T ), (2.35)

∂tV − ∂xV + V

∫ 1

0

U dy −∫ 1

0

W 2 dy = 0 in R × (0, T ), (2.36)

∂tW +W 2 −∫ 1

0

U(x, y − z, t)V (x, y + z, t) dz = 0 in R × (0, T ), (2.37)


with initial values given by

U(x, y, 0) = u0(x, y), V (x, y, 0) = v0(x, y), W (x, y, 0) = w0(x, y). (2.38)

Note that U(x, y, t), V (x, y, t),W (x, y, t) are 1-periodic in y and the system (2.35)-(2.38) is a set of partialdifferential equations in (x, t) with y ∈ [0, 1] as a parameter. The global existence of the systems (2.31)-(2.34)and (2.35)-(2.38) has been established, see the references cited in [49].

Theorem 2.1. Let (u, v, w) and (U, V,W ) be the solutions of the systems (2.31)-(2.34) and (2.35)-(2.38),respectively. Then we have the following error estimate

max0≤t≤T

E(t) ≤[

5(M(T )2 + 2TK(T )M(T )) exp(6M(T )T )]

ε := C1(T )ε, (2.39)

where the error function E(t) is given by

E(t) = maxx∈R

∣

∣

∣u(x, t) − U(x, x−tε , t)

∣

∣

∣+∣

∣

∣v(x, t) − V (x, x+tε , t)

∣

∣

∣

+∣

∣

∣w(x, t) −W (x, xε , t)

∣

∣

∣

and the constants M(T ) and K(T ) are given by

M(T ) = max(x,y,t)∈R×[0,1]×[0,T ]

(

|u|, |v|, |w|, |U |, |V |, |W |)

, (2.40)

N(T ) = max(x,y,t)∈R×[0,1]×[0,T ]

(

|∂xU |, |∂tU |, |∂xV |, |∂tV |, |∂xW |, |∂tW |)

. (2.41)

This homogenization result was first obtained by McLaughlin, Papanicolaou and Tartar using an Lp normestimate (0 < p < ∞) [81]. Since we need an L∞ norm estimate in the convergence analysis of our particlemethod, we give another proof of this result in L∞ norm. As a first step, we prove the following lemma.

Lemma 2.3. Let g(x, y) ∈ C1(R × [0, 1]) be 1-periodic in y and satisfy the relation∫ 1

0 g(x, y) dy = 0. Thenfor any ε > 0 and for any constants a and b, the following estimate holds

∣

∣

∣

∫ b

a

g(x, xε ) dx∣

∣

∣≤ B(g)ε+ |b− a|B(∂xg)ε, (2.42)

where B(ζ) = max(x,y)∈R×[0,1] |ζ(x, y)| for any function ζ defined on R × [0, 1].

Proof. The estimate (2.42) is a direct consequence of the identity

g(x, xε ) =d

dx

∫ x

a

g(x, sε ) ds−∫ x

a

∂g

∂x(x, sε ) ds

and the estimates∣

∣

∣

∫ b

a

g(x, sε ) ds∣

∣

∣≤ B(g)ε,

∣

∣

∣

∫ x

a

∂g

∂x(x, sε ) ds

∣

∣

∣≤ B(∂xg)ε,

which follow from the 1-periodicity of g(x, y) in y and that∫ 1

0 g(x, y) dy = 0. This completes the proof. ut

10 Thomas Y. Hou

Proof (of Theorem 2.1). Subtracting (2.35) from (2.31) and integrating the resulting equation along thecharacteristics from 0 to t, we get

u(x, t) − U(x, x−tε , t)

=

∫ t

0

[

w(x − t+ s, s)2 −W (x− t+ s, x−t+sε , s)2]

ds

+

∫ t

0

[

W (x− t+ s, x−t+sε , s)2 −∫ 1

0 W (x− t+ s, y, s)2 dy]

ds

−∫ t

0

[

u(x− t+ s, s)v(x− t+ s, s)

−U(x− t+ s, x−tε , s)V (x − t+ s, x−t+2sε , s)

]

ds

−∫ t

0

U(x− t+ s, x−tε , s)[

V (x − t+ s, x−t+2sε , s)

−∫ 1

0

V (x− t+ s, y, s) dy]

ds

:= (I)1 + · · · + (I)4. (2.43)

It is clear from the definition of E(t) and M(T ) that

|(I)1 + (I)3| ≤ 2M(T )

∫ t

0

E(s) ds.

To estimate (I)2, we define for fixed (x, t) ∈ R × [0, T ],

g(x,t)(s, y) = W (x− t+ s, x−tε + y, s)2.

Since the 1-periodicity of W (x, y, t) in y implies

∫ 1

0

W (x− t+ s, y, s)2 dy =

∫ 1

0

W (x− t+ s, x−tε + y, s)2 dy,

we obtain by applying Lemma 2.1 that

|(I)2| =∣

∣

∣

∫ t

0

[

g(x,t)(s,sε ) −

∫ 1

0

g(x,t)(s, y) dy]

ds∣

∣

∣

≤ M(T )2ε+ 2M(T )K(T )Tε.

Similarly, we have(I)4 ≤M(T )2ε+ 2M(T )K(T )Tε.

Substituting these estimates into (2.43) we get

∣

∣

∣u(x, t) − U(x, x−tε , t)

∣

∣

∣≤ 2M(T )

∫ t

0

E(s) ds+ 2M(T )2ε+ 4M(T )K(T )Tε.

(2.44)

Similarly, we conclude from (2.36)-(2.37) and (2.32)-(2.33) that

∣

∣

∣v(x, t) − V (x, x+tε , t)

∣

∣

∣≤ 2M(T )

∫ t

0

E(s) ds+ 2M(T )2ε+ 4M(T )K(T )Tε,(2.45)

∣

∣

∣w(x, t) −W (x, xε , t)

∣

∣

∣≤ 2M(T )

∫ t

0

E(s) ds+M(T )2ε+ 2M(T )K(T )Tε.(2.46)

Now the desired estimate (2.39) follows from summing (2.44)-(2.46) and using the Gronwall inequality. ut


Remark 2.2. The homogenization theory tells us that the initial oscillatory solutions propagate along theircharacteristics. The nonlinear interaction can generate only low frequency contributions to the u and v com-ponents. On the other hand, the nonlinear interaction of u, v on w can generate both low and high frequencycontribution to w. That is, even if w has no oscillatory component initially, the dynamical interaction of u, vand w can generate a high frequency contribution to w at later time. This is not the case for the u and vcomponents. Due to this resonant interaction of u, v and w, the weak limit of uεvεwε is not equal to theproduct of the weak limits of uε, vε, wε. This explains why the Compensated Compactness result does notapply to this 3 × 3 system [98].

Although it is difficult to characterize the weak limit of the triple product, uεvεwε for arbitrary oscillatoryinitial data, it is possible to say something about the weak limit of the triple product for oscillatory initialdata that have periodic structure, such as the ones studies here. Depending on α being rational or irrational,the limiting behavior is very different. In fact, one can show that uεvεwε = uεvεwε when α is equal to anirrational number. This is not true in general when α is a rational number.

2.3 Convection of microstructure

It is most interesting to see if one can apply homogenization technique to obtain an averaged equation for thelarge scale quantity for incompressible Euler or Navier-Stokes equations. In 1985, McLaughlin, Papanicolaouand Pironneau [82] attempted to obtain a homogenized equation for the 3-D incompressible Euler equationswith highly oscillatory velocity field. More specifically, they considered the following initial value problem:

ut + (u · ∇)u = −∇p,

with ∇ · u = 0 and highly oscillatory initial data

u(x, 0) = U(x) +W (x, x/ε).

They then constructed multiscale expansions for both the velocity field and the pressure. In doing so, theymade an important assumption that the microstructure is convected by the mean flow. Under this assumption,they constructed a multiscale expansion for the velocity field as follows:

uε(x, t) = u(x, t) + w( θ(x,t)ε , tε , x, t) + εu1(θ(x,t)ε , tε , x, t) +O(ε2).

The pressure field pε is expanded similarly. From this ansatz, one can show that θ is convected by the meanvelocity:

θt + u · ∇θ = 0 , θ(x, 0) = x .

It is a very challenging problems to develop a systematic approach to study the large scale solution inthree dimensional Euler and Navier-Stokes equations. The work of McLaughlin, Papanicolaou and Pironneauprovided some insightful understanding into how small scales interact with large scale and how to deal withthe closure problem. However, the problem is still not completely resolved since the cell problem obtainedthis way does not have a unique solution. Additional constraints need to be enforced in order to derive alarge scale averaged equation. With additional assumptions, they managed to derive a variant of the k − εmodel in turbulence modeling.

Remark 2.3. One possible way to improve the work of [82] is take into account the oscillation in the La-grangian characteristics, θε. The oscillatory part of θε in general could have order one contribution to themean velocity of the incompressible Euler equation. In [65–67], Hou and Yang and co-workers have studiedconvection of microstructure of the 2-D and 3-D incompressible Euler equations using a new approach. Theydo not assume that the oscillation is propagated by the mean flow. In fact, they found that it is crucial toinclude the effect of oscillations in the characteristics on the mean flow. Using this new approach, they canderive a well-posed cell problem which can be used to obtain an effective large scale average equation.

12 Thomas Y. Hou

More can be said for a passive scalar convection equation.

vt +1

ε∇ · (u(x/ε)v) = α∆v,

with v(x, 0) = v0(x). Here u(y) is a known incompressible periodic (or stationary random) velocity field withzero mean. Assume that the initial condition is smooth.

Expand the solution vε in powers of ε

vε = v(t, x) + εv1(t, x, x/ε) + ε2v2(t, x, x/ε) + · · · .The coefficients of ε−1 lead to

α∆yv1 − u · ∇yv1 − u · ∇xv = 0.

Let ek, k = 1, 2, 3 be the unit vectors in the coordinate directions and let χk(y) satisfy the cell problem:

α∆yχk − u · ∇yχ

k − u · ek = 0.

Then we have

v1(t, x, y) =

3∑

k=1

χk(y)v(t, x)

∂xk.

The coefficients of ε0 give

α∆yv2 − u · ∇yv2 = u · ∇xv1 − 2α∇x · ∇yv1 − α∆xv + vt.

The solvability condition for v2 requires that the right hand side has zero mean with respect to y. This givesrise to the equation for homogenized solution v

vt = α∆xv − u · ∇xv1.

Using the cell problem, McLaughlin, Papanicolaou, and Pironneau obtained [82]

vt =

3∑

i,j=1

(αδij + αTij )∂2v

∂xi∂xj,

where αTij = −uiχj .

Nonlocal memory effect of homogenization It is interesting to note that for certain degenerate problem,the homogenized equation may have a nonlocal memory effect.

Consider the simple 2-D linear convection equation:

∂uε(x, y, t)

∂t+ aε(y)

∂uε(x, y, t)

∂x= 0,

with initial condition uε(x, y, 0) = u0(x, y). Note that y = x2 is not a fast variable here.We assume that aε is bounded and u0 has compact support. While it is easy to write down the solution

explicitly,uε(x, y, t) = u0(x− aε(y)t, y),

it is not an easy task to derive the homogenized equation for the weak limit of uε.Using Laplace Transform and measure theory, Luc Tartar [99] showed that the weak limit u of uε satisfies

∂

∂tu(x, y, t) +A1(y)

∂

∂xu(x, y, t) =

∫ t

0

∫

∂2

∂x2u(x− λ(t− s), y, s)dµy(λ) ds,

with u(x, y, 0) = u0(x, y), where A1(y) is the weak limit of aε(y), and µy is a probability measure of y andhas support in [min(aε),max(aε)].

As we can see, the degenerate convection induces a nonlocal history dependent diffusion term in thepropagating direction (x). The homogenized equation is not amenable to computation since the measure µycannot be expressed explicitly in terms of aε.


3 Numerical Homogenization Based on Sampling Techniques

Homogenization theory provides a critical guideline for us to design effective numerical methods to computemultiscale problems. Whenever homogenized equations are applicable they are very useful for computationalpurposes. There are, however, many situations for which we do not have well-posed effective equations or forwhich the solution contains different frequencies such that effective equations are not practical. In these caseswe would like to approximate the original equations directly. In this part of my lectures, we will investigate thepossibility of approximating multiscale problems using particle methods together with sampling technique.The classes of equations we consider here include semilinear hyperbolic systems and the incompressible Eulerequation with oscillatory solutions.

When we talk about convergence of an approximation to an oscillatory solution, we need to introducea new definition. The traditional convergence concept is too weak in practice and does not discriminatebetween solutions which are highly oscillatory and those which are smooth. We need the error to be smallessentially independent of the wavelength in the oscillation when the computational grid size is small. On theother hand we cannot expect the approximation to be well behaved pointwise. It is enough if the continuoussolution and its discrete approximation have similar local or moving averages.

Definition 3.1 (Engquist [48]). Let vn be the numerical approximation to u at time tn(tn = n∆t), εrepresents the wave length of oscillation in the solution. The approximation vn converges to u as ∆t → 0,essentially independent of ε, if for any δ > 0 and T > 0 there exists a set s(ε,∆t0) ∈ (0,∆t0) with measure(s(ε,∆t0)) ≥ (1 − δ)∆t0 such that

||u(·, tn) − vn|| ≤ δ, 0 ≤ tn ≤ T

is valid for all ∆t ∈ s(ε,∆t0) and where ∆t0 is independent of ε.

The convergence concept of “essentially independent of ε” is strong enough to mimic the practical casewhere the high frequency oscillations are not well resolved on the grid. A small set of values of ∆t has tobe removed in order to avoid resonance between ∆t and ε. Compare the almost always convergence for theMonte Carlo methods [86].

It is natural to compare our problem with the numerical approximation of discontinuous solutions ofnonlinear conservation laws. Shock capturing methods do not produce the correct shock profiles but theoverall solution may still be good. For this the scheme must satisfy certain conditions such as conservationform. We are here interested in analogous conditions on algorithms for oscillatory solutions. These conditionsshould ideally guarantee that the numerical approximation in some sense is close to the solution of thecorresponding effective equation when the wave length of the oscillation tends to zero.

There are three central sources of problems for discrete approximations of highly oscillatory solutions.

(i) The first one is the sampling of the computational mesh points (xj = j∆x, j = 0, 1, ...). There is therisk of resonance between the mesh points and the oscillation. For example, if ∆x equals the wavelength of the periodic oscillation, the discrete initial data may only get values from the peaks of a curvelike the upper envelope of the oscillatory solution. We can never expect convergence in that case. Thus∆x cannot be completely independent of the wave length.

(ii) Another problem comes from the approximation of advection. The group velocity for the differentialequation and the corresponding discretization are often very different [51]. This means that an oscilla-tory pulse which is not well resolved is not transported correctly even in average by the approximation.Furthermore, dissipative schemes do not advect oscillations correctly. The oscillations are damped outvery fast in time.

(iii) Finally, the nonlinear interaction of different high frequency components in a solution must be mod-eled correctly. High frequency interactions may produce lower frequencies that influence the averagedsolution. We can show that this nonlinear interaction is well approximated by certain particle methodsapplied to a class of semilinear differential equations. The problem is open for the approximation ofmore general nonlinear equations.

14 Thomas Y. Hou

In [50,49], we studied a particle method approximation to the nonlinear discrete Boltzmann equations inkinetic theory of discrete velocity with multiscale initial data. In such equations, high frequency componentscan be transformed into lower frequencies through nonlinear interactions, thus affecting the average of solu-tions. We assume that the initial data are of the form a(x, x/ε) with a(x, y) 1-periodic in each componentof y. As we see from the homogenization theory in the previous section, the behavior of oscillatory solutionsfor the generalized Broadwell model is very sensitive to the velocity coefficients. It depends on whether acertain ratio among the velocity components is a rational number or an irrational number.

It is interesting to note that the structure of oscillatory solutions for the generalized Broadwell model isquite stable when we perturb the velocity coefficient α around irrational numbers. In this case, the resonanceeffect of u and v on w vanishes in the limit of ε → 0. However, the behavior of oscillatory solutions for thegeneralized Broadwell model becomes singular when perturbing around integer velocity coefficients. There isa strong interaction between the high frequency components of u and v, and the interaction in the uv termwould create an oscillation of order O(1) on the w component. In [98], Tartar showed that for the Carlemanmodel the weak limit of all powers of the initial data will uniquely determine the weak limit of the oscillatorysolutions at later times, using the Compensated Compactness Theorem. We found that this is no longer truefor the generalized Broadwell model with integer-values velocity coefficients [59].

In [50,49], we showed that this subtle behavior for the generalized Broadwell model with oscillatory initialdata can be captured correctly by a particle method even on a coarse grid. The particle method convergesto the effective solution essentially independent of ε. For the Broadwell model, the hyperbolic part is solvedexactly by the particle method. No averaging is therefore needed in the convergence result. We also analyzea numerical approximation of the Carleman equations with variable coefficients. The scheme is designedsuch that particle interaction can be accounted for without introducing interpolation. There are errors inthe particle method approximation of the linear part of the system. As a result, the convergence can onlybe proved for moving averages. The convergence proofs for the Carleman and the Broadwell equations haveone feature in common. The local truncation errors in both cases are of order O(∆t). In order to showconvergence, we need to take into account cancellation of the local errors at different time levels. This isvery different from the conventional convergence analysis for finite difference methods. This is also the placewhere numerical sampling becomes crucial in order to obtain error cancellation at different time levels.

In the next two subsections, we present a careful study of the Broadwell model with highly oscillatoryinitial data in order to demonstrate the basic idea of the numerical homogenization based on samplingtechniques.

3.1 Convergence of the Particle Method

Now we consider how to capture this oscillatory solution on a coarse grid using a particle method. Since thediscrete velocity coefficients are integers for the Broadwell model, we can express a particle method in theform of a special finite difference method by choosing ∆x = ∆t. Denote by uni , v

ni , w

ni the approximations of

u(xi, tn), v(xi, t

n) and w(xi, tn) respectively with xi = i∆x and tn = n∆t. Our particle scheme is given by

uni = un−1i−1 + ∆t(w2 − uv)n−1

i−1 , (3.1)

vni = vn−1i+1 + ∆t(w2 − uv)n−1

i+1 , (3.2)

wni = wn−1i − ∆t(w2 − uv)n−1

i , (3.3)

with the initial conditions given by

u0i = u(xi, 0), v0

i = v(xi, 0), w0i = w(xi, 0). (3.4)

To study the convergence of the particle scheme (3.1)-(3.4) we need the following lemma, which is a discreteanalogue of Lemma 2.1.


Lemma 3.1. Let g(x, y) ∈ C3([0, T ]× [0, 1]) be 1-periodic in y and satisfy the relation∫ 1

0g(x, y) dy = 0. Let

xk = kh and r = h/ε. If h ∈ S(ε, h0) where

S(ε, h0) = 0 < h ≤ h0 :kh

ε6∈(

i− τ

|k|3/2 , i+τ

|k|3/2)

,

for i = 1, 2, · · · ,[kh0

ε

]

+ 1, 0 6= k ∈ Z, 0 < ε ≤ 1,

then we have∣

∣

∣

n−1∑

k=0

g(xk,xkε

)h∣

∣

∣≤ C0(1 + T )L(g)h

τ, ∀ n = 1, 2, · · · ,

[T

h

]

,

where C0 is a constant independent of h, ε, T, τ and g, and

L(g) = max(x,y)∈[0,T ]×[0,1]

(

|∂3yg(x, y)|, |∂x∂3

yg(x, y)|)

.

Moreover, it is obvious that

|S(ε, h0)| ≥ h0

(

1 − τ∞∑

k=1

k−3/2)

≥ h(1 − 3τ).

Proof. Since g is 1-periodic in y with mean zero, it can be expanded in a Fourier series

g(x, y) =∑

m6=0

am(x)e2πimy , where am(x) =

∫ 1

0

g(x, y)e−2πimy dy.

Simple integration by parts yields that

|am(x)| ≤ 1

(2π|m|)3L(g), |a′m(x)| ≤ 1

(2π|m|)3L(g).

Thus we have

∣

∣

∣

n−1∑

k=0

g(xk,xkε

)h∣

∣

∣=∣

∣

∣

n−1∑

k=0

∑

m6=0

am(xk)e2πimxk/εh

∣

∣

∣

=∣

∣

∣

∑

m6=0

n−1∑

k=0

am(xk)e2πimkh/εh

∣

∣

∣.

Summation by parts yields

∣

∣

∣

n−1∑

k=0

am(xk)e2πikh/ε

∣

∣

∣≤ |am(xn−1)

n−1∑

k=0

e2πikh/ε∣

∣

∣

+∣

∣

∣

n−1∑

k=0

(

k∑

j=1

e2πimjh/ε)

(am(xk) − am(xk+1))∣

∣

∣

≤ 2(1 + T )L(g)

(2π|m|)3|1 − e2πimh/ε| .

But for h ∈ S(ε, h0) we have

|1 − e2πimh/ε| = 2| sin(πmh/ε)| ≥ 2πτ

|m|3/2 .

16 Thomas Y. Hou

Hence, for h ∈ S(ε, h0),

n−1∑

k=0

∣

∣

∣g(xk,

xkε

)h∣

∣

∣≤ 2(1 + T )L(g)h

(2π)4τ

∑

m6=0

1

|m|3/2 =:C0h(1 + T )L(g)

τ.

This completes the proof. utNow we are ready to study the approximation property of the particle scheme (3.1)-(3.4). First denote

by

En = maxi

(

|u(xi, tn) − uni |, |v(xi, tn) − vni |, |w(xi, tn) − wni |

)

. (3.5)

Integrating (2.28) from 0 to tn along its characteristics, we get

u(xi, tn) = u(xi − tn, 0) +

∫ tn

0

(w2 − uv)(xi − tn + s, s) ds. (3.6)

From (3.1) we know that

uni = u0i−n +

n−1∑

k=0

(w2 − uv)ki−k∆t. (3.7)

Subtracting (3.7) from (3.6) we obtain that

u(xi, tn) − uni

=

∫ tn

0

(w2 − uv)(xi − tn + s, s) ds−n−1∑

k=0

(w2 − uv)(xi − tk, tk)∆t

+

n−1∑

k=0

∆t[

(w2 − uv)(xi − tk, tk) − (w2 − uv)ki−k

]

:= (II) + (III). (3.8)

Let M(T ) be defined as in (2.40) and N(T ) be given by

N(T ) = max

|uki |, |vki |, |wki | : i ∈ Z, 0 ≤ k ≤ [T/∆t]

. (3.9)

It can be shown that N(T ) is bounded for finite time independent of ε, see [49]. Then it is clear that

(III) ≤ (M(T ) +N(T ))

n−1∑

k=0

∆tEk. (3.10)

It remains to estimate (II). For convenience, let θ = w2 − uv and

Θ(x, t) = W (x, xε , t)2 − U(x, x−tε , t)V (x, x+tε , t).

Then we have

(II) =

∫ tn

0

[

θ(xi − tn + s, s) ds− Θ(xi − tn + s, s)]

ds

+[

∫ tn

0

Θ(xi − tn + s, s) ds−n−1∑

k=0

∆tΘ(xi − tk, tk)]

+

n−1∑

k=0

[

Θ(xi − tk, tk) − θ(xi − tk, tk)]

∆t

:= (II)1 + · · · + (II)3. (3.11)


By Theorem 2.1 we get

|(II)1 + (II)3| ≤ 2TM(T )C1(T )ε. (3.12)

To proceed further, let, for fixed (xi, tn),

gni (s, y) = W (xi − tn + s, xi−tn

ε + y, s)2.

It is clear that gni is 1-periodic in y. Now by Lemmas 2.1-2.2 we have

∫ tn

0

W (xi − tn + s, s)2 ds−n−1∑

k=0

W (xi − tk, tk)2∆t

=

∫ tn

0

[

gni (s, sε ) −∫ 1

0

gni (s, y) dy]

ds

+

∫ tn

0

∫ 1

0

gni (s, y) dy ds−n−1∑

k=0

∆t

∫ 1

0

gni (tn−k, y) dy

−n−1∑

k=0

∆t[

g(tn−k, tn−k

ε ) −∫ 1

0

gni (tn−k, y) dy]

≤ C(T )(ε+ ∆t), (3.13)

where we have used standard methods to estimate the second term, since the derivative of gni with respect tos is independent of ε. Here and in the remainder of this section, we will always denote by C(T ) the variousconstants which are independent of ε and ∆t. Now similar to the reasoning leading to (3.13) we can obtain

|(II)3| ≤ C(T )(ε+ ∆t). (3.14)

From (3.8)-(3.12) and (3.14) we finally get

|u(xi, tn) − uni | ≤ C(T )(ε+ ∆t) + (M(T ) +N(T ))

n−1∑

k=0

∆tEk . (3.15)

Similarly, we have

|v(xi, tn) − vni | ≤ C(T )(ε+ ∆t) + (M(T ) +N(T ))

n−1∑

k=0

∆tEk, (3.16)

|w(xi, tn) − wni | ≤ C(T )(ε+ ∆t) + (M(T ) +N(T ))

n−1∑

k=0

∆tEk . (3.17)

To summarize, we have the following theorem by summing (3.15)-(3.17) and applying the Gronwall inequality.

Theorem 3.1. Let (u, v, w) be the solution of (2.31)-(2.34) and (uni , vni , w

ni ) be the solution of the particle

scheme (3.1)-(3.4). Assume that ∆t ∈ S(ε,∆t0) where S(ε,∆t0) is defined in Lemma 3.1. Then the followingestimate holds

max1≤n≤[T/∆t]

En ≤ C(T )(ε+ ∆t),

where C(T ) is independent of ε and ∆t, and En is defined as in (3.5).

Remark 3.1. It is important that we perform the error analysis globally in time in order to account forcancellation of local truncation errors at different time steps. As we can see from the analysis, the localtruncation error is of order ∆t in one time step. If we do not take into account the error cancellation in time,we would obtain an error bound of order O(1) which is an over-estimate. The error cancellation is closelyrelated to the sampling we choose. This is the place where we can see the difference between a good samplingand a resonant sampling.

18 Thomas Y. Hou

Remark 3.2. As we can see from the error analysis, error cancellation along Lagrangian characteristics isessential in obtaining convergence independent of the oscillation. This idea can be generalized to hyperbolicsystems with variable coefficient velocity fields. In the special case of the Carleman model with variablecoefficients, we have analyzed the convergence of a particle method in [50]. However, the particle methodanalyzed in [50] does not generalize to multi-dimensions or 3 × 3 systems. Together Razvan Fetecau [54],we have designed a modified Lagrangian particle method. In this method, each component of the solutionis updated along its own characteristic. So there is no fixed grid. When we update one component of thesolution, say u, we need values of the other components (say v and w) along the u characteristic. Weobtain these values by using some high order interpolation scheme (such as cubic spline). In the case of theCarleman model with variable coefficient and the non-resonant Broadwell model (α being irrational), we canprove rigorously the convergence of the modified particle method essentially independent of the small scale.Our numerical experiments show that the modified particle method works even for the original Broadwellmodel, which is surprising. This modified Lagrangian particle method in principle works for any number offamilies of characteristics and for multi-dimensions.

Below we describe briefly the results we obtain for the variable coefficient Carleman equations

ut + a(x, t)ux = v2 − u2 , (3.18)

vt − b(x, t)vx = u2 − v2 , (3.19)

with initial data u(x, 0) = u0(x, x/ε), v(x, 0) = v0(x, x/ε). In Figure 3.1, we illustrate the particle trajectoriesfor the u and v components.

Fig. 3.1. Schematic particle trajectories for different components.

We choose the oscillatory coefficients as follows:

a(x, t) = 1 + 0.5 sin(

xtε

)

and b(x, t) = 1 + 0.2 cos(

xtε

)

.

The initial conditions for u and v are chosen as

u0(x, x/ε) =

0.5 sin4(π(x − 3)/2)(1 + sin(2π(x− 3)/ε)),0,

|x− 4| < 1|x− 4| ≥ 1

(3.20)

v0(x, x/ε) =

0.5 sin4(π(x − 4)/2)(1 + sin(2π(x− 4)/ε)),0,

|x− 5| < 1|x− 5| ≥ 1

(3.21)

In our calculations, we choose ∆x = 0.01, ∆t = ∆x√5, and ε = ∆x

√2 ≈ 0.014. We plot the u-characteristic

in Figure 3.2. The coarse grid solution for the u-component is plotted in Figure 3.3a. We can see that itcaptures very well the high frequency information. In Figure 3.3b, we put the coarse grid solution on topof the corresponding well-resolved solution. The agreement is very good. We also check the accuracy of the


moving average [50] of the solution and the average of its second order moments. The results are plotted inFigure 3.4. Again, we observe excellent agreement between the coarse grid calculations and the well-resolvedcalculations.

We have also performed the same calculations for the 3 × 3 Broadwell model with rational or irrationalcoefficient α. The subtle homogenization behavior is captured correctly for both rational α and for irrationalα. We do not present the results here.

3.6 3.65 3.7 3.75 3.8 3.85 3.90

0.05

0.1

0.15

0.2

0.25

0.3

Fig. 3.2. A typical u-characteristic trajectory.

4 4.5 5 5.5 60

0.2

0.4

0.6

0.8

4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2 5.30

0.2

0.4

0.6

0.8

Fig. 3.3. (a): Coarse grid solution u at time t = 1.28. (b): Putting the coarse grid solution u on top of a well-resolvedcomputation (solid line).

3.2 Vortex methods for incompressible flows

The generalization of the particle method to the incompressible flows is the vortex method. In [38], wehave analyzed the convergence of the vortex method for 2-D incompressible Euler equations with oscillatoryvorticity field. Our analysis relies on the observation that there are tremendous cancellations among thelocal errors at different space locations in the velocity approximation. Thus the local errors do not add upto O(1) as predicted by the classical error estimate in the case where the grid size is large compared to theoscillatory wavelength.

Consider the 2-D incompressible Euler equation in vorticity form:

ωt + (u · ∇)ω = 0

20 Thomas Y. Hou

3 3.5 4 4.5 5 5.5 6 6.5 70

0.1

0.2

0.3

0.4

0.5

0.6

3 3.5 4 4.5 5 5.5 6 6.5 70

0.1

0.2

0.3

0.4

0.5

0.6

Fig. 3.4. (a): The averaged solution u (dashdot line); the solid line represents a very well resolved computation. (b):The averaged second order moment u2 (dashdot line); the solid line represents a very well resolved computation.

with oscillatory initial vorticity ω(x, 0) = ω0(x, x/ε).Define the particle trajectory, denoted as X(t, α),

dX(t, α)

dt= u(X(t, α), t), X(0, α) = α.

Vorticity is conserved along characteristics:

ω(X(t, α), t) = ω0(α).

On the other hand, velocity can be expressed in terms of vorticity by the Biot-Savart law:

u(X(t, α), t) =

∫

K(X(t, α) −X(t, α′))ω0(α′)dα′

with K given by K(x) = (−x2, x1)/(2π|x|2).The Biot-Savart kernel K has a singularity at the origin. To regularize the kernel, Chorin introduced the

vortex blob method (see, e.g. [26], replacing K by Kδ = K ∗ ζδ ,

ζδ =1

δ2ζ(x

δ

)

, δ = hσ, with σ < 1.

ζ is typically chosen as a variant of Gaussian.The vortex blob method is given by

dXhi (t)

dt=∑

j

Kδ(Xhi (t) −Xh

j (t))ωjh2,

where Xhi (0) = αi, and wj = w0(αj , αj/ε).

Together with Weinan E, we have proved that the vortex method converges essentially independent of ε[38].

The case studied in [38] deals with bounded oscillatory vorticity. This assumption leads to strong con-vergence of the velocity field. It is more physical to consider homogenization for highly oscillatory velocityfield. Would the vortex blob method still capture the correct large scale solution with a relatively coarse grid(or small number of particles)? This is still an open question.

4 Numerical Upscaling based on Multiscale Finite Element Methods

It is natural to consider the possibility of generalizing the sampling technique to second order elliptic equa-tions with highly oscillatory coefficients. In [8], we showed that finite difference approximations converge


essentially independent of the small scale ε for one-dimensional elliptic problems. In several space dimen-sions we found that only in the case of rapidly oscillating periodic coefficients do the above results generalize,in a weaker form. In the case of almost periodic or random coefficients in several space dimensions we showed,both theoretically and with a simple counterexample, that numerical homogenization by sampling does notwork efficiently. New ideas seem to be needed.

In order to overcome the difficulty we mentioned above for the sampling technique, we have introduced amultiscale finite element method (MsFEM) for solving partial differential equations with multiscale solutions,see [62,64,63,42,25,100,3,40]. The central goal of this approach is to obtain the large scale solutions accuratelyand efficiently without resolving the small scale details. The main idea is to construct finite element basefunctions which capture the small scale information within each element. The small scale information is thenbrought to the large scales through the coupling of the global stiffness matrix. Thus, the effect of small scaleson the large scales is correctly captured. In our method, the base functions are constructed from the leadingorder homogeneous elliptic equation in each element. As a consequence, the base functions are adapted tothe local microstructure of the differential operator. In the case of two-scale periodic structures, we haveproved that the multiscale method indeed converges to the correct solution independent of the small scalein the homogenization limit [64].

In practical computations, a large amount of overhead time comes from constructing the base functions.In general, these multiscale base functions are constructed numerically, except for certain special cases. Sincethe base functions are independent of each other, they can be constructed independently and can be doneperfectly in parallel. This greatly reduces the overhead time in constructing these bases. In many applications,it is important to obtain a scale-up equation from the fine grid equation. For example, the high degree ofvariability and multiscale nature of formation properties in subsurface flows (such as permeability) posesignificant challenges for subsurface flow modeling. Geological characterizations that capture these effectsare typically developed at scales that are too fine for direct flow simulation, so techniques are required toenable the solution of flow problems in practice. Upscaling procedures have been commonly applied for thispurpose and are effective in many cases (see e.g., [72] for reviews and discussion). Our multiscale finiteelement method can be used for a similar purpose and successfully applied for problems of this type.

As discussed in [72], upscaling methods and multiscale numerical techniques (as applied within the contextof subsurface flow modeling) have many similarities and some important differences. Upscaling techniquesprovide coefficients, which are typically computed in a pre-processing step, for coarse scale equations ofprescribed analytical forms. In multiscale methods, the coarse scale equations are formed numerically andfine scale information may be carried throughout the simulation and used at various stages. For example, inmultiscale procedures for subsurface flow applications, different grids are often used for flow and transportcomputations. The advantage of deriving a scale-up equation or performing multiscale computations is thatone can perform many useful tests on the coarse model with different boundary conditions or source terms.This would be very expensive if we have to perform all these tests on a fine grid. For time dependent problems,the coarse-scale equation also allows for larger time steps. This results in additional computational saving.

It should be mentioned that many numerical methods have been developed with goals similar to ours.These include generalized finite element methods [13,11,10], wavelet based numerical homogenization meth-ods [18,31,29,76], methods based on the homogenization theory (cf. [16,35,28,52]), equation-free computa-tions (e.g., [75]), variational multiscale methods [70,21,71], heterogeneous multiscale methods [37], matrix-dependent multigrid based homogenization [76,29], generalized p-FEM in homogenization [78,79], and someupscaling methods based on simple physical and/or mathematical motivations (cf. [33,80]). The methodsbased on the homogenization theory have been successfully applied to determine the effective conductivityand permeability of certain composite materials and porous media. However, their range of applications isusually limited by restrictive assumptions on the media, such as scale separation and periodicity [15,74].They are also expensive to use for solving problems with many separate scales since the cost of computationgrows exponentially with the number of scales. But for the multiscale method, the number of scales doesnot increase the overall computational cost exponentially. The upscaling methods are more general and havebeen applied to problems with random coefficients with partial success (cf. [33,80]). But the design principleis strongly motivated by the homogenization theory for periodic structures. Their application to nonperiodicstructures is not always guaranteed to work.

22 Thomas Y. Hou

Most multiscale methods presented to date have applied local calculations for the determination of basisfunctions. Though effective in many cases, global effects can be important for some problems. The importanceof global information has been illustrated within the context of upscaling procedures as well as multiscalecomputations in recent investigations. These studies have shown that the use of limited global information inthe calculation of the coarse-scale parameters (such as basis functions) can significantly improve the accuracyof the resulting coarse model. In this lecture notes, we describe the use of limited global information inmultiscale simulations.

We remark that the idea of using base functions governed by the differential equations has been appliedto convection-diffusion equation with boundary layers (see, e.g., [14] and references therein). Babuska et al.applied a similar idea to 1-D problems [13] and to a special class of 2-D problems with the coefficient varyinglocally in one direction [11]. Most of these methods are based on the special property of one-dimensionalproperties of the coefficients. As indicated by our convergence analysis, there is a fundamental differencebetween one-dimensional problems and genuinely multi-dimensional problems. Special complications such asthe resonance between the mesh scale and the physical scale never occur in the corresponding 1-D problems.

4.1 Multiscale Finite Element Methods for Elliptic PDEs.

In this section we consider the multiscale finite element method applied to the following problem

Lεp := −∇ · (a(xε )∇p) = f in Ω, p = 0 on Γ = ∂Ω, (4.1)

where Ω is a convex polygon in R2. The unknown is changed to p, since it will be used later in porous media

flow simulations, where the solution represents the pressure field. ε is assumed to be a small parameter, anda(x) = (aij(x/ε)) is symmetric and satisfies α|ξ|2 ≤ aijξiξj ≤ β|ξ|2, for all ξ ∈ R

2 and with 0 < α < β.Furthermore, aij(y) are smooth periodic function in y in a unit cube Y . We will always assume that f ∈L2(Ω). In fact, the smoothness assumption on aij can be relaxed, which will be discussed later.

Let p0 be the solution of the homogenized equation

L0p0 := −∇ · (a∗∇p0) = f in Ω, p0 = 0 on Γ, (4.2)

where Γ = ∂Ω and

a∗ij =1

|Y |

∫

Y

aik(y)(δkj −∂χj

∂yk) dy,

and χj(y) is the periodic solution of the cell problem

∇y · (a(y)∇yχj) =

∂

∂yiaij(y) in Y,

∫

Y

χj(y) dy = 0.

It is clear that p0 ∈ H2(Ω) since Ω is a convex polygon. Denote by p1(x, y) = −χj(y)∂p0(x)∂xj

and let θε be the

solution of the problem

Lεθε = 0 in Ω, θε(x) = p1(x,xε ) on Γ. (4.3)

Our analysis of the multiscale finite element method relies on the following homogenization result obtainedby Moskow and Vogelius [83].

Lemma 4.1. Let p0 ∈ H2(Ω) be the solution of (4.2), θε ∈ H1(Ω) be the solution to (4.3) and p1(x) =−χj(x/ε)∂p0(x)/∂xj . Then there exists a constant C independent of u0, ε and Ω such that

‖ p− p0 − ε(u1 − θε) ‖1,Ω ≤ Cε(| p0 |2,Ω + ‖ f ‖0,Ω).

Now we are going to introduce the multiscale finite element methods. Let Th be a regular partition ofΩ into triangles. Let xjJj=1 be the interior nodes of the mesh Th and ψjJj=1 be the nodal basis of the


standard linear finite element space Wh ⊂ H10 (Ω). Denote by Si = supp(ψi) and define φi with support in

Si as follows:

Lεφi = 0 in K, φi = ψi on ∂K ∀ K ∈ Th,K ⊂ Si. (4.4)

It is obvious that φi ∈ H10 (Si) ⊂ H1

0 (Ω). Finally, let Vh ⊂ H10 (Ω) be the finite element space spanned by

φiJi=1.With above notation we can introduce the following discrete problem: find ph ∈ Vh such that

(a(xε )∇ph,∇vh) = (f, vh) ∀ vh ∈ Vh, (4.5)

where and hereafter we denote by (·, ·) the L2 inner product in L2(Ω).As we will see later, the choice of boundary conditions in defining the multiscale bases will play a crucial

role in approximating the multiscale solution. Intuitively, the boundary condition for the multiscale basefunction should reflect the multiscale oscillation of the solution p across the boundary of the coarse gridelement. By choosing a linear boundary condition for the base function, we will create a mismatch betweenthe exact solution p and the finite element approximation across the element boundary. In the next section,we will discuss this issue further and introduce an over-sampling technique to alleviate this difficulty. Theover-sampling technique plays an important role when we need to reconstruct the local fine grid velocityfield from a coarse grid pressure computation for two-phase flows. This technique enables us to remove theartificial numerical boundary layer across the coarse grid boundary element.

We remark that the multiscale finite element method with linear boundary conditions for the multiscalebase functions is similar in spirit to the residual-free bubbles finite element method [20] and the variationalmultiscale method [70,21]. In a recent paper [93], Dr. G. Sangalli derives a multiscale method based on theresidual-free bubbles formulation in [93] and compares it with the multiscale finite element method describedhere. There are many striking similarities between the two approaches.

To gain some insight into the multiscale finite element method, we next perform an error analysis forthe multiscale finite element method in the simplest case, i.e. we use linear boundary conditions for themultiscale base functions.

4.2 Error Estimates (h < ε)

The starting point is the well-known Cea’s lemma.

Lemma 4.2. Let p be the solution of (4.1) and ph be the solution of (4.5). Then we have

‖ p− ph ‖1,Ω ≤ C infvh∈Vh

‖ p− vh ‖1,Ω.

Let Πh : C(Ω) →Wh ⊂ H10 (Ω) be the usual Lagrange interpolation operator:

Πhp(x) =J∑

j=1

p(xj)ψj(x) ∀ u ∈ C(Ω)

and Ih : C(Ω) → Vh be the corresponding interpolation operator defined through the multiscale base functionφ

Ihp(x) =

J∑

j=1

p(xj)φj(x) ∀ u ∈ C(Ω).

From the definition of the basis function φi in (4.4) we have

Lε(Ihp) = 0 in K, Ihp = Πhp on ∂K, (4.6)

for any K ∈ Th.

24 Thomas Y. Hou

Lemma 4.3. Let p ∈ H2(Ω) be the solution of (4.1). Then there exists a constant C independent of h, εsuch that

‖ p− Ihp ‖0,Ω + h‖ p− Ihp ‖1,Ω ≤ Ch2(| p |2,Ω + ‖ f ‖0,Ω). (4.7)

Proof. At first it is known from the standard finite element interpolation theory that

‖ p− Πhp ‖0,Ω + h‖ p− Πhp ‖1,Ω ≤ Ch2(| p |2,Ω + ‖ f ‖0,Ω). (4.8)

On the other hand, since Πhp− Ihp = 0 on ∂K, the standard scaling argument yields

‖Πhp− Ihp ‖0,K ≤ Ch|Πhp− Ihp|1,K ∀ K ∈ Th. (4.9)

To estimate |Πhp− Ihp|1,K we multiply the equation in (4.6) by Ihp− Πhp ∈ H10 (K) to get

(a(xε )∇Ihp,∇(Ihp− Πhp))K = 0,

where (·, ·)K denotes the L2 inner product of L2(K). Thus, upon using the equation in (4.1), we get

(a(xε )∇(Ihp− Πhp),∇(Ihp− Πhp))K

= (a(xε )∇(p− Πhp),∇(Ihp− Πhp))K − (a(xε )∇p,∇(Ihp− Πhp))K

= (a(xε )∇(p− Πhp),∇(Ihp− Πhp))K − (f, Ihp− Πhp)K .

This implies that

|Ihp− Πhp|1,K ≤ Ch| p |2,K + ‖ Ihp− Πhp ‖0,K‖ f ‖0,K .

Hence

|Ihp− Πhp|1,K ≤ Ch(| p |2,K + ‖ f ‖0,K), (4.10)

where we have used (4.9). Now the lemma follows from (4.8)-(4.10). ut

In conclusion, we have the following estimate by using Lemmas 4.2-4.3.

Theorem 4.1. Let p ∈ H2(Ω) be the solution of (4.1) and ph ∈ Vh be the solution of (4.5). Then we have

‖ p− ph ‖1,Ω ≤ Ch(| p |2,Ω + ‖ f ‖0,Ω). (4.11)

Note that the estimate (4.11) blows up like h/ε as ε → 0 since | p |2,Ω = O(1/ε). This is insufficient forpractical applications. In next subsection we derive an error estimate which is uniform as ε→ 0.

4.3 Error Estimates (h > ε)

In this section, we will show that the multiscale finite element method gives a convergence result uniform inε as ε tends to zero. This is the main feature of this multiscale finite element method over the traditionalfinite element method. The main result in this subsection is the following theorem.

Theorem 4.2. Let p ∈ H2(Ω) be the solution of (4.1) and ph ∈ Vh be the solution of (4.5). Then we have

‖ p− ph ‖1,Ω ≤ C(h+ ε)‖ f ‖0,Ω + C( ε

h

)1/2

‖ p0 ‖1,∞,Ω, (4.12)

where p0 ∈ H2(Ω) ∩W 1,∞(Ω) is the solution of the homogenized equation (4.2).


To prove the theorem, we first denote by

pI(x) = Ihp0(x) =J∑

j=1

p0(xj)φj(x) ∈ Vh.

From (4.6) we know that LεpI = 0 in K and pI = Πhp0 on ∂K for any K ∈ Th. The homogenization theory(see (2.26)) implies that

‖ pI − pI0 − ε(pI1 − θIε) ‖1,K ≤ Cε(‖ f ‖0,K + | pI0 |2,K), (4.13)

where pI0 is the solution of the homogenized equation on K:

L0pI0 = 0 in K, pI0 = Πhp0 on ∂K, (4.14)

pI1 is given by the relation

pI1(x, y) = −χj(y)∂pI0

∂xjin K, (4.15)

and θIε ∈ H1(K) is the solution of the problem:

LεθIε = 0 in K, θIε(x) = pI1(x,xε ) on ∂K. (4.16)

It is obvious from (4.14) that

pI0 = Πhp0 in K, (4.17)

since Πhp0 is linear on K. From (4.13) we obtain that

‖ p− pI ‖1,Ω ≤ ‖ p0 − pI0 ‖1,Ω + ‖ ε(p1 − pI1) ‖1,Ω

+‖ ε(θε − θIε) ‖1,Ω + Cε‖ f ‖0,Ω, (4.18)

where we have used the regularity estimate ‖ p0 ‖2,Ω ≤ C‖ f ‖0,Ω. Now it remains to estimate the terms atthe right-hand side of (4.18).

Lemma 4.4. We have

‖ p0 − pI0 ‖1,Ω ≤ Ch‖ f ‖0,Ω, (4.19)

‖ ε(p1 − pI1) ‖1,Ω ≤ C(h+ ε)‖ f ‖0,Ω. (4.20)

Proof. The estimate (4.19) is a direct consequence of the standard finite element interpolation theory sincepI0 = Πhp0 by (4.17). Next we note that χj(x/ε) satisfies

‖χj ‖0,∞,Ω + ε‖∇χj ‖0,∞,Ω ≤ C (4.21)

for some constant C independent of h and ε. Thus we have, for any K ∈ Th,

‖ ε(p1 − pI1) ‖0,K ≤ Cε‖χj ∂

∂xj(p0 − Πhp0) ‖0,K ≤ Chε| p0 |2,K ,

‖ ε∇(p1 − pI1) ‖0,K = ε‖∇(χj∂(p0 − Πhp0)

∂xj) ‖0,K

≤ C‖∇(p0 − Πhp0) ‖0,K + Cε| p0 |2,K≤ C(h+ ε)| p0 |2,K .

This completes the proof. ut

26 Thomas Y. Hou

Lemma 4.5. We have

‖ εθε ‖1,Ω ≤ C√ε‖ p0 ‖1,∞,Ω + Cε| p0 |2,Ω. (4.22)

Proof. Let ζ ∈ C∞0 (R2) be the cut-off function which satisfies ζ ≡ 1 in Ω\Ωδ/2, ζ ≡ 0 in Ωδ, 0 ≤ ζ ≤ 1 in

R2, and |∇ζ| ≤ C/δ in Ω, where for any δ > 0 sufficiently small, we denote by Ωδ as

Ωδ = x ∈ Ω : dist(x, ∂Ω) ≥ δ.

With this definition, it is clear that θε − ζp1 = θε + ζ(χj∂p0/∂xj) ∈ H10 (Ω). Multiplying the equation in

(4.3) by θε − ζp1, we get

(a(xε )∇θε,∇(θε + ζχj∂p0

∂xj)) = 0,

which yields, by using (4.21),

‖∇θε ‖0,Ω ≤ C‖∇(ζχj∂p0/∂xj) ‖0,Ω

≤ C‖∇ζ · χj∂p0/∂xj ‖0,Ω + C‖ ζ∇χj∂p0/∂xj ‖0,Ω

+C‖ ζχj∂2p0/∂2xj ‖0,Ω

≤ C√

|∂Ω| · δDδ

+ C√

|∂Ω| · δDε

+ C| p0 |2,Ω, (4.23)

where D = ‖ p0 ‖1,∞,Ω and the constant C is independent of the domain Ω. From (4.23) we have

‖ εθε ‖0,Ω ≤ C(ε√δ

+√δ)‖ p0 ‖1,∞,Ω + Cε| p0 |2,Ω

≤ C√ε‖ p0 ‖1,∞,Ω + Cε| p0 |2,Ω. (4.24)

Moreover, by applying the maximum principle to (4.3), we get

‖ θε ‖0,∞,Ω ≤ ‖χj∂p0/∂xj ‖0,∞,∂Ω ≤ C‖ p0 ‖1,∞,Ω. (4.25)

Combining (4.24) and (4.25) completes the proof. ut

Lemma 4.6. We have

‖ εθIε ‖1,Ω ≤ C( ε

h

)1/2

‖ p0 ‖1,∞,Ω. (4.26)

Proof. First we remember that for any K ∈ Th, θIε ∈ H1(K) satisfies

LεθIε = 0 in K, θIε = −χj(xε)∂(Πhp0)

∂xjon ∂K. (4.27)

By applying maximum principle and (4.21) we get

‖ θIε ‖0,∞,K ≤ ‖χj∂(Πhp0)/∂xj ‖0,∞,∂K ≤ C‖ p0 ‖1,∞,K .

Thus we have

‖ εθIε ‖0,Ω ≤ Cε‖ p0 ‖1,∞,Ω. (4.28)

On the other hand, since the constant C in (4.23) is independent of Ω, we can apply the same argumentleading to (4.23) to obtain

‖ ε∇θIε ‖0,K ≤ Cε‖Πhp0 ‖1,∞,K(√

|∂K|/√δ +

√

|∂K| · δ/ε) + Cε|Πhp0 |2,K≤ C

√h‖ p0 ‖1,∞,K(

ε√δ

+√δ)

≤ C√hε‖ p0 ‖1,∞,K ,


which implies that

‖ ε∇θIε ‖0,Ω ≤ C( ε

h

)1/2

‖ p0 ‖1,∞,Ω.

This completes the proof. utProof (of Theorem 3.2.). The theorem is now a direct consequence of (4.18) and the Lemma 4.4-4.6 and theregularity estimate ‖ p0 ‖2,Ω ≤ C‖ f ‖0,Ω. utRemark 4.1. As we pointed out earlier, the multiscale FEM indeed gives correct homogenized result as εtends to zero. This is in contrast with the traditional FEM which does not give the correct homogenizedresult as ε → 0. The error would grow like O(h2/ε2). On the other hand, we also observe that when h ∼ ε,the multiscale method attains large error in both H1 and L2 norms. This is what we call the resonance effectbetween the grid scale (h) and the small scale (ε) of the problem. This estimate reflects the intrinsic scaleinteraction between the two scales in the discrete problem. Our extensive numerical experiments confirmthat this estimate is indeed generic and sharp. From the viewpoint of practical applications, it is importantto reduce or completely remove the resonance error for problems with many scales since the chance of hittinga resonance sampling is high. In the next subsection, we propose an over-sampling method to overcome thisdifficulty.

4.4 The Over-Sampling Technique

As illustrated by our error analysis, large errors result from the “resonance” between the grid scale andthe scales of the continuous problem. For the two-scale problem, the error due to the resonance manifestsas a ratio between the wavelength of the small scale oscillation and the grid size; the error becomes largewhen the two scales are close. A deeper analysis shows that the boundary layer in the first order correctorseems to be the main source of the resonance effect. By a judicious choice of boundary conditions for the basefunction, we can eliminate the boundary layer in the first order corrector. This would give a nice conservativedifference structure in the discretization, which in turn leads to cancellation of resonance errors and givesan improved rate of convergence.

Motivated by our convergence analysis, we propose an over-sampling method to overcome the difficultydue to scale resonance [62]. The idea is quite simple and easy to implement. Since the boundary layer in thefirst order corrector is thin, O(ε), we can sample in a domain with size larger than h + ε and use only theinterior sampled information to construct the bases; here, h is the mesh size and ε is the small scale in thesolution. By doing this, we can reduce the influence of the boundary layer in the larger sample domain onthe base functions significantly. As a consequence, we obtain an improved rate of convergence.

Specifically, let ψj be the base functions satisfying the homogeneous elliptic equation in the larger domainS ⊃ K. We then form the actual base φi by linear combination of ψj ,

φi =

d∑

j=1

cijψj .

The coefficients cij are determined by condition φi(xj) = δij . The corresponding θiε for φi are now free ofboundary layers. Our extensive numerical experiments have demonstrated that the over-sampling techniquedoes improve the numerical error substantially in many applications. On the other hand, the over-samplingtechnique results in a non-conforming MsFEM method. In [42], we perform a careful estimate of the non-conforming errors in both H1 norm and the L2 norm. The analysis shows that the non-conforming erroris indeed small, consistent with our numerical results [62,63]. Our analysis also reveals another source ofresonance, which is the mismatch between the mesh size and the “perfect” sample size. In case of a pe-riodic structure, the “perfect” sample size is the length of an integer multiple of the period. We call thenew resonance the “cell resonance”. In the error expansion, this resonance effect appears as a higher ordercorrection. In numerical computations, we found that the cell resonance error is generically small, and israrely observed in practice. Nonetheless, it is possible to completely eliminate this cell resonance error byusing the over-sampling technique to construct the base functions but using piecewise linear functions as testfunctions. This reduces the nonconforming error and eliminates the resonance error completely (see [60]).

28 Thomas Y. Hou

4.5 Performance and Implementation Issues

The multiscale method given in the previous section is fairly straightforward to implement. Here, we outlinethe implementation and define some notations that are used in the discussion below. We consider solvingproblems in a unit square domain. Let N be the number of elements in the x and y directions. The mesh sizeis thus h = 1/N . To compute the base functions, each element is discretized into M ×M subcell elementswith mesh size hs = h/M . To implement the over-sampling method, we partition the domain into samplingdomains and each of them contains many elements. From the analysis and numerical tests, the size of thesampling domains can be chosen freely as long as the boundary layer is avoided. In practice, though, onewants to maximize the efficiency of over-sampling by choosing the largest possible sample size which reducesthe redundant computation of overlapping domains to a minimum.

In general, the multiscale (sampling) base functions are constructed numerically, except for certain specialcases. They are solved in each K or S using standard FEM. The linear systems are solved using a robustmultigrid method with matrix dependent prolongation and ILLU smoothing (MG-ILLU, see [101]). Theglobal linear system on Ω is solved using the same method. Numerical tests show that the accuracy of thefinal solution is insensitive to the accuracy of base functions.

Since the base functions are independent of each other, their construction can be carried out in parallelperfectly. In our parallel implementation of over-sampling, the sample domains are chosen such that they canbe handled within each processor without communication. The multigrid solver is also modified to bettersuit the parallelization. In particular, the ILLU smoothing is replaced by Gauss-Seidel iterations. Moreimplementation details can be found in [62].

Cost and Performance In practical computations, a large amount of overhead time comes from construct-ing the base functions. On a sequential machine, the operation count of our method is about twice that ofa conventional FEM for a 2-D problem. However, due to good parallel efficiency, this difference is reducedsignificantly on a massively parallel computer. For example, using 256 processors on an Intel Paragon, ourmethod with N = 32 and M = 32 only spends 9% more CPU time than the conventional linear FEM methodusing 1024 × 1024 elements [62]. Note that this comparison is made for a single solve of the problem. Inpractice, multiple solves are often required, then the overhead of base construction is negligible. A detailedstudy of MsFEM’s parallel efficiency has been conducted in [62]. It was also found that MsFEM is helpfulfor improving multigrid convergence when the coefficient aε has very large contrast (i.e., the ratio betweenthe maximum and minimum of aε).

Significant computational savings can be obtained for time dependent problems (such as two-phase flows)by constructing the multiscale bases adaptively. Multiscale base functions are updated only for those coarsegrid elements where the saturation changes significantly. In practice, the number of such coarse grid elementsare small. They are concentrated near the interface separating oil and water. Also, the cost of solving a basefunction in a small cell is more efficient than solving the fine grid problem globally because the conditionnumber for solving the local base function in each coarse grid element is much smaller than that of thecorresponding global fine grid pressure system. Thus, updating a small number of multiscale base functionsdynamically is much cheaper than updating the fine grid pressure field globally.

Another advantage of the multiscale finite element method is its ability to scale down the size of a largescale problem. This offers a big saving in computer memory. For example, let N be the number of elementsin each spatial direction, and M be the number of subcell elements in each direction for solving the basefunctions. Then there are total (MN)n (n is dimension) elements at the fine grid level. For a traditionalFEM, the computer memory needed for solving the problem on the fine grid is O(MnNn). In contrast,MsFEM requires only O(Mn +Nn) amount of memory. For a typical value of M = 32 in a 2-D problem,the traditional FEM needs about 1000 times more memory than MsFEM.

MsFEM for problems with scale separation If there is a scale separation in representative volumessmaller than the coarse block, then multiscale finite element basis functions can be computed based on thesmaller regions. To demonstrate this, we first consider a periodic case. In this case, the basis functions can


be approximated by

φj(x) = φj0(x) + εχi∇iφj0.

Consequently, the approximation of the basis functions can be carried out in a domain of size ε via thecomputation of χi. This reduces the computational cost. Moreover, the assembly of stiffness matrix can bealso performed in a period, because a(x/ε)∇φi · ∇φj is a periodic function. The results obtained by thisapproximation gives the classical numerical homogenization procedure that is based on the computation ofeffective coefficients based on periodic problems. We would like to note that this approximation procedureis not limited to periodic problems and can be applied to random homogeneous problems with the strongscale separation, i.e., the size of representative volume is much smaller than the coarse mesh size. In general,this holds for problems where homogenization by periodization (see [74]) is true. Random homogeneous casewith ergodicity is one of them. We note that a number of methods used in practice employs this strategy(e.g., [92,75,55,37]).

Convergence and Accuracy Since we need to use an additional grid to compute the base functionnumerically, it makes sense to compare our MsFEM with a traditional FEM at the subcell grid, hs = h/M .Note that MsFEM only captures the solution at the coarse grid h, while FEM tries to resolve the solutionat the fine grid hs. Our extensive numerical experiments demonstrate that the accuracy of MsFEM on thecoarse grid h is comparable to that of FEM on the fine grid. In some cases, MsFEM is even more accuratethan the FEM (see below and the next section).

As an example, in Table 4.1 we present the result for

a(x/ε) =2 + P sin(2πx/ε)

2 + P cos(2πy/ε)+

2 + sin(2πy/ε)

2 + P sin(2πx/ε)(P = 1.8), (4.29)

f(x) = −1 and u|∂Ω = 0. (4.30)

The convergence of three different methods are compared for fixed ε/h = 0.64, where “-L” indicates thatlinear boundary condition is imposed on the multiscale base functions, “os” indicates the use of over-sampling,and LFEM stands for standard FEM with linear base functions. We see clearly the scale resonance in the

MsFEM-L MsFEM-os-L LFEMN ε

||E||l2 rate ||E||l2 rate MN ||E||l2

16 0.04 3.54e-4 7.78e-5 256 1.34e-432 0.02 3.90e-4 -0.14 3.83e-5 1.02 512 1.34e-464 0.01 4.04e-4 -0.05 1.97e-5 0.96 1024 1.34e-4128 0.005 4.10e-4 -0.02 1.03e-5 0.94 2048 1.34e-4

Table 4.1. Convergence for periodic case.

results of MsFEM-L and the (almost) first order convergence (i.e., no resonance) in MsFEM-os-L. Evidentalso is the error of MsFEM-os-L being smaller than those of LFEM obtained on the fine grid. In [64,62],more extensive convergence tests have been presented.

4.6 Applications

Flow in Porous Media One of the main application of our multiscale method is the flow and transportthrough porous media. This is a fundamental problem in hydrology and petroleum engineering. Here, weapply MsFEM to solve the single phase flow, which is a good test problem in practice.

We model the porous media by random distributions of aε generated using a spectral method. In fact,aε = α10βp, where p is a random field represents porosity, and α and β are scaling constants to give thedesired contrast of aε. In particular, we have tested the method for a porous medium with a statisticallyfractal porosity field (see Figure 4.1). The fractal dimension is 2.8. This is a model of flow in an oil reservoir

30 Thomas Y. Hou

Fig. 4.1. Porosity field with fractal dimension of 2.8 generated using the spectral method.

or aquifer with uniform injection in the domain and outflow at the boundaries. We note that the problemhas a continuous scale because of the fractal distribution.

The pressure field due to uniform injection is solved and the error is shown in Figure 4.2. The horizontaldash line indicates the error of the LFEM solution withN = 2048. The coarse-grid solutions are obtained withdifferent number of elements, N , but fixed NM = 2048. We note that error of MsFEM-os-L almost coincide

1e-4

5e-4

1e-3

32 64 128 256 512

L2-n

orm

Err

or

N

LFEMMFEM-OMFEM-L

MFEM-os-L

Fig. 4.2. The l2-norm error of the solutions using various schemes for a fractal distributed permeability field.

with that of the well-resolved solution obtained using LFEM. However, MsFEM without over-sampling is lessaccurate. MsFEM-O indicates that oscillatory boundary conditions, obtained from solving some reduced 1-Delliptic equations along ∂K (see [62]), are imposed on the base functions. The decay of error in MsFEM isbecause of the decay of small scales in aε. The next figure shows the results for a log-normally distributed aε.In this case, the effect of scale resonance shows clearly for MsFEM-L, i.e., the error increases as h approachesε. Here ε ∼ 0.004 roughly equals the correlation length. Using the oscillatory boundary conditions (MsFEM-O) gives better results, but it does not completely eliminate resonance. On the other hand, the multiscalemethod with over-sampling agrees extremely well with the well-resolved calculation. One may wonder whythe errors do not decrease as the number of coarse grid elements increase. This is because we use the samesubgrid mesh size, which is the same as the well-resolved grid size, to construct the base functions for variouscoarse grid sizes (N = 32, 64, 128, etc). In some special cases, one can construct multiscale base functions


analytically. In this case, the errors for the coarse grid computations will indeed decrease as the number ofcoarse grid elements increase.

5e-4

1e-3

5e-3

1e-2

32 64 128 256 512

L2-n

orm

Err

or

N

LFEMMFEM-OMFEM-L

MFEM-os-L

Fig. 4.3. The l2-norm error of the solutions using various schemes for a log-normally distributed permeability field.

Fine Scale Recovery To solve transport problems in the subsurface formations, as in oil reservoir simula-tions, one needs to compute the velocity field from the elliptic equation for pressure, i.e v = −aε∇u, here u ispressure. In some applications involving isotropic media, the cell-averaged velocity is sufficient, as shown bysome computations using the local upscaling methods (cf. [33]). However, for anisotropic media, especiallylayered ones (Figure 4.4), the velocity in some thin channels can be much higher than the cell average, andthese channels often have dominant effects on the transport solutions. In this case, the information aboutfine scale velocity becomes vitally important. Therefore, an important question for all upscaling methods ishow to take those fast-flow channels into account.

Fig. 4.4. A random porosity field with layered structure.

For MsFEM, the fine scale velocity can be easily recovered from the multiscale base functions, noting thatthey provide interpolations from the coarse h-grid to the fine hs-grid. Using the over-sampling technique,the error in velocity is O(ε/h), as proved in [42]. We remark that the resonance effect seems unavoidable

32 Thomas Y. Hou

in the velocity. On the other hand, our numerical tests indicate that the error is small when ε ≈ h. Thecell-averaged velocity can also be obtained and its error is even smaller.

Fig. 4.5. (a): Fine grid horizontal velocity field, N = 1024. (b): Recovered horizontal velocity field from the coarsegrid N = 64 calculation using multiscale bases.

Fig. 4.6. (a): Fine grid saturation at t = 0.06, N = 1024. (b): Saturation computed using the recovered velocityfield from the coarse grid calculation.

To demonstrate the accuracy of the recovered velocity and effect of small-scale velocity on the transportproblem, we show the fractional flow result of a “tracer” test using the layered medium in Figure 4.4: afluid with red color originally saturating the medium is displaced by the same fluid with blue color injectedby flow in the medium at the left boundary, where the flow is created by a unit horizontal pressure drop.The linear convection equation is solved to compute the saturation of the red fluid (for details, see [34]). Todemonstrate that we can recover the fine grid velocity field from the coarse grid pressure calculation, we plotthe horizontal velocity fields obtained by two methods. In Figure 4.5a, we plot the horizontal velocity fieldobtained by using a fine grid (N = 1024) calculation. In Figure 4.5b, we plot the same horizontal velocityfield obtained by using the coarse grid pressure calculation with N = 64 and using the multiscale finite


element bases to interpolate the fine grid velocity field. We can see that the recovered velocity field capturesvery well the layer structure in the fine grid velocity field. Further, we use the recovered fine grid velocityfield to compute the saturation in time. In Figure 4.6a, we plot the saturation at t = 0.06 obtained by thefine grid calculation. Figure 4.6b shows the corresponding saturation obtained using the recovered velocityfield from the coarse grid calculation. The agreement is striking.

We also check the fractional flow curves obtained by the two calculations. The fractional flow of thered fluid, defined as F =

∫

Sredvx dy/∫

vx dy (S being the saturation), at the right boundary is shown inFigure 4.7. The top pair of curves are the solutions of the transport problem using the cell-averaged velocityobtained from a well-resolved solution and from MsFEM; the bottom pair are solutions using well-resolvedfine scale velocity and the recovered fine scale velocity from the MsFEM calculation. Two conclusions can bemade from the comparisons. First, the cell-averaged velocity may lead to a large error in the solution of thetransport equation. Second, both recovered fine scale velocity and the cell-averaged velocity obtained fromMsFEM give faithful reproductions of respective direct numerical solutions.

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6

Fra

ctio

nal F

low

Time

DNS (fine)MFEM (recovered)

DNS (averaged)MFEM (coarse)

Fig. 4.7. Variation of fractional flow with time. DNS: well-resolved direct numerical solution using LFEM (N = 512).MsFEM: over-sampling is used (N = 64, M = 8).

Scale-up of one-phase flows The multiscale finite element method has been used in conjunction withsome moment closure models to obtain an upscaled method for one-phase flows, see, e.g. [39,46,25]. Notethat the multiscale finite element method presented above does not conserve mass. For long time integration,it may lead to significant loss of mass. This is an undesirable feature of the method. In a recent work withZhiming Chen [25], we have designed and analyzed a mixed multiscale finite element method, and we haveapplied this mixed method to study the scale up of one-phase flows and found that mass is conserved verywell even for long time integration. Below we describe our results in some detail.

In its simplest form, neglecting the effect of gravity, compressibility, capillary pressure, and consideringconstant porosity and unit mobility, the governing equations for the flow transport in highly heterogeneousporous media can be described by the following partial differential equations [77], [102], and [39]

div(k(x)∇p) = 0, (4.31)

∂S

∂t+ v · ∇S = 0, (4.32)

where p is the pressure, S is the water saturation, k(x) = (kij(x)) is the relative permeability tensor, andv = −k(x)∇p is the Darcy velocity. The highly heterogeneous properties of the medium are built into the

34 Thomas Y. Hou

permeability tensor k(x) which is generated through the use of sophisticated geological and geostatisticalmodeling tools. The detailed structure of the permeability coefficients makes the direct simulation of theabove model infeasible. For example, it is common in real simulations to use millions of grid blocks, witheach block having a dimension of tens of meters, whereas the permeability measured from cores is at a scaleof centimeters [80]. This gives more than 105 degrees of freedom per spatial dimension in the computa-tion. This makes a direct simulation to resolve all small scales prohibitive even with today’s most powerfulsupercomputers. On the other hand, from an engineering perspective, it is often sufficient to predict themacroscopic properties of the solutions. Thus it is highly desirable to derive effective coarse grid models tocapture the correct large scale solution without resolving the small scale features. Numerical upscaling is oneof the commonly used approaches in practice.

Now we describe how the (mixed) multiscale finite element can be combined with the existing upscalingtechnique for the saturation equation (4.32) to get a complete coarse grid algorithm for the problem (4.31)-(4.32). The numerical upscaling of the saturation equation has been under intensive study in the literature[34,46,77,58,105,103]. Here, we use the upscaling method proposed in [46] and [39] to design an overall coarsegrid model for the problem (4.31)-(4.32). The work of [46] for upscaling the saturation equation involves amoment closure argument. The velocity and the saturation are separated into a local mean quantity anda small scale perturbation with zero mean. For example, the Darcy velocity is expressed as v = v0 + v′

in (4.32), where v0 is the average of velocity v over each coarse element, v′ = (v′1,v

′2) is the deviation of

the fine scale velocity from its coarse scale average. After some manipulations, an average equation for thesaturation S can be derived as follows [46]:

∂S

∂t+ v0 · ∇S =

∂

∂xi

(

Dij(x, t)∂S

∂xj

)

, (4.33)

where the diffusion coefficients Dij(x, t) are defined by

Dii(x, t) = 〈|v′i(x)|〉L0

i (x, t), Dij(x, t) = 0, for i 6= j,

〈|v′i(x)|〉 stands for the average of |v′

i(x)| over each coarse element. L0i (x, t) is the length of the coarse grid

streamline in the xi direction which starts at time t at point x, i.e.

L0i (x, t) =

∫ t

0

yi(s) ds,

where y(s) is the solution of the following system of ODEs

dy(s)

ds= v0(y(s)), y(t) = x.

Note that the hyperbolic equation (4.32) is now replaced by a convection-diffusion equation. The convection-dominant parabolic equation (4.33) is solved by the characteristic linear finite element method [32], [91] inour simulation. The flow transport model (4.31)-(4.32) is solved in the coarse grid as follows:

1. Solve the pressure equation (4.31) by the over-sampling mixed multiscale finite element method andobtain the fine scale velocity field using the multiscale basis functions.

2. Compute the coarse grid average v0 and the fine scale deviation 〈|v′i(x)|〉 on the coarse grid.

3. At each time step, solve the convection-diffusion equation (4.33) by the characteristic linear finite elementmethod on the coarse grid in which the lengths L0

i (x, t) of the streamline are computed for the center ofeach coarse grid element.

The mixed multiscale finite element method can be readily combined with the above upscaling modelfor the saturation equation. The local fine grid velocity v′ will be constructed from the multiscale finiteelement base functions. The main cost in the above algorithm lies in the computation of multiscale baseswhich can be done a priori and completely in parallel. This algorithm is particularly attractive when multiple


simulations must be carried out due to the change of boundary and source distribution as it is often thecase in engineering applications. In such a situation, the cost of computing the multiscale base functions isjust an over-head. Moreover, once these base functions are computed, they can be used for subsequent timeintegration of the saturation. Because the evolution equation is now solved on a coarse grid, a larger timestep can be used. This also offers additional computational saving. For many oil recovery problems, due tothe excessively large fine grid data, upscaling is a necessary step before performing many simulations andrealizations on the upscaled coarse grid model. If one can coarsen the fine grid by a factor of 10 in eachdimension, the computational saving of the coarse grid model over the original fine model could be as largeas a factor 10,000 (three space dimensions plus time).

We perform a coarse grid computation of the above algorithm on the coarse 64×64 mesh. The fractionalflow curve using the above algorithm is depicted in Figure 4.8. It gives excellent agreement with the “exact”fractional flow curve. The contour plots of the saturation S on the fine 1024 × 1024 mesh at time t = 0.25and t = 0.5 computed by using the “exact” velocity field are displayed in Figure 4.10. In Figure 4.9, we showthe contour plots of the saturation obtained using the recovered velocity field from the coarse grid pressurecalculation N = 64. We can see that the the contour plots in Figure 4.9 approximate the “exact” ones inFigure 4.10 in certain accuracy but the sharp oil/water interfaces in Figure 4.10 are smeared out. This isdue to the parabolic nature of the upscaled equation (4.33). We have also performed many other numericalexperiments to test the robustness of this combined coarse grid model. We found that for permeability fieldswith strong layered structure, the above coarse grid model is very robust. The agreement with the fine gridcalculations is very good. We are currently working toward some qualitative and quantitative understandingof this upscaling model.

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

t

F(t)

Fig. 4.8. The accuracy of the coarse grid algorithm. Solid line is the “exact” fractional flow curve using mixed finiteelement method solving the pressure equation. The slash-dotted line is the fractional flow curve using above coarsegrid algorithm.

Finally, we remark that the upscaling equation (4.33) uses small scale information v′ of the velocity fieldto define the diffusion coefficients. This information can be constructed locally through the mixed multiscalebasis functions. This is an important property of our multiscale finite element method. It is clear that solvingdirectly the homogenized pressure equation

div(k∗(x)∇p∗) = 0

will not provide such small scale information. On the other hand, whenever one can afford to resolve allthe small scale feature by a fine grid, one can use fast linear solvers, such as multigrid methods, to solvethe pressure equation (4.31) on the fine mesh. From the fine grid computation, one can easily construct theaverage velocity v0 and its deviation v′. However, when multiple simulations must be carried out due to the

36 Thomas Y. Hou

10 20 30 40 50 60

10

20

30

40

50

60

(a)

10 20 30 40 50 60

10

20

30

40

50

60

(b)

Fig. 4.9. The contour plots of the saturation S computed using the upscaled model on a 64 × 64 mesh at timet = 0.25 (left) and t = 0.5 (right).

100 200 300 400 500 600 700 800 900 1000

100

200

300

400

500

600

700

800

900

1000

(a)

100 200 300 400 500 600 700 800 900 1000

100

200

300

400

500

600

700

800

900

1000

(a)

Fig. 4.10. The contour plots of the saturation S computed on the fine 1024 × 1024 mesh using the “exact” velocityfield at time t = 0.25 (left) and t = 0.5 (right).

change of boundary conditions, the pressure equation (4.31) will then have to be solved again on the finemesh. The multiscale finite element method only solves the pressure equation once on a coarse mesh, andthe fine grid velocity can be constructed locally through the finite element bases. This is the main advantageof our mixed multiscale finite element method. This process becomes more difficult for the nonlinear two-phase flow due to the dynamic coupling between the pressure and the saturation. We are now investigatingthe possibility of upscaling the two-phase flow by using multiscale finite element base functions that areconstructed from the one-phase flow (time independent). In this case, we need to provide corrections to thepressure equation to account for the scale interaction near the oil/water interface.

It should be noted that some adaptive scale-up strategies have also been developed [34,105]. The ideais to refine the mesh around the fast-flow channels in order to capture their effect directly. The approachseems to work well when the channels are isolated. For MsFEM, it is also possible to adjust the coarse meshadaptively based on the recovered velocity. In particular, one does not need to use the fine recovered velocityin the regions with no fast-flow channels; in those regions, the coarse mesh and cell-averaged velocity aresufficient. On the other hand, one can simply keep the fine mesh when the channels are too many. How todevelop a consistent upscaling equation for the saturation equation is still open when the capillary pressureeffect is neglected, which is the common practice in oil reservoir simulations. One approach is to combinegrid adaptivity with multiscale modeling. We use a dynamic adaptive coarse grid [24] to capture the isolatedsmall scale features, such as the flow channels and use the multiscale finite element method to capture thesmall scale feature within each adaptive coarse grid block. By doing this, we take into account the local floworientation and anisotropy in upscaling the saturation equation. We are also investigating the possibilityto develop a consistent upscaling model for the saturation equation by combining multiscale finite elementmethods and systematic multiscale modeling for the saturation equation.


4.7 Brief overview of mixed finite element and finite volume element methods

Control volume multiscale finite element method In this section, we discuss multiscale finite volumeelement method. Finite volume method is chosen because, by its construction, it satisfies the numerical localconservation which is important in groundwater and reservoir simulations. Let Kh denote the collection ofcoarse elements/rectangles K. Consider a coarse element K, and let ξK be its center. The element K isdivided into four rectangles of equal area by connecting ξK to the midpoints of the element’s edges. Wedenote these quadrilaterals by Kξ, where ξ ∈ Zh(K), are the vertices of K. Also, we denote Zh =

⋃

K Zh(K)and Z0

h ⊂ Zh the vertices which do not lie on the Dirichlet boundary of Ω. The control volume Vξ is definedas the union of the quadrilaterals Kξ sharing the vertex ξ.

The key idea of the method is the construction of basis functions on the coarse grids, such that these basisfunctions capture the small-scale information on each of these coarse grids. As before, the basis functionsare constructed from the solution of the leading order homogeneous elliptic equation on each coarse elementwith some specified boundary conditions. We consider a coarse element K that has d vertices, the local basisfunctions φi, i = 1, · · · , d are set to satisfy the following elliptic problem:

−∇ · (k · ∇φi) = 0 inK

φi = gi on ∂K,(4.34)

for some function gi defined on the boundary of the coarse element K. As we discussed earlier, Hou et al. [62]have demonstrated that a careful choice of boundary conditions would improve the accuracy of the method.In previous findings, the function gi for each i is chosen to vary linearly along ∂K or to be the solution ofthe local one-dimensional problems [73] or the solution of the problem in a slightly larger domain is chosento define the boundary conditions. For simplicity, we consider linear boundary conditions and also discussthe boundary conditions obtained from a global solution. We will require φi(xj) = δij . Finally, a nodal basisfunction associated with the vertex xi in the domain Ω is constructed from the combination of the localbasis functions that share this xi and zero elsewhere. We would like to note that one can use an approximatesolution of (4.34) when it is possible. For example, in the case of periodic or random homogeneous cases, thebasis functions can be approximated using homogenization expansion φi = φ0

i + εχk∇kφ0i , where χk is the

solution of the cell problem and φ0i is standard finite element basis on the coarse mesh (see [41]).

Next, we denote by V h the space of our approximate pressure solution, which is spanned by the basisfunctions φjxj∈Z0

h. Then we formulate the finite dimensional problem corresponding to finite volume el-

ement formulation of pressure equation. A statement of mass conservation on a coarse-control volume Vxis formed from pressure equation, where the approximate solution is written as a linear combination of thebasis functions. Assembly of this conservation statement for all control volumes would give the correspondinglinear system of equations that can be solved accordingly. The resulting linear system has incorporated thefine-scale information through the involvement of the nodal basis functions on the approximate solution. Tobe specific, the problem now is to seek ph ∈ V h with ph =

∑

xj∈Z0hpjφj such that

∫

∂Vξ

k · ∇ph · n dl = 0, (4.35)

for every control volume Vξ ⊂ Ω. Here n defines the normal vector on the boundary of the control volume,∂Vξ and S is the fine-scale saturation field at this point. The resulting multiscale method differs from themultiscale finite element method, since it employs the finite volume element method as a global solver, andit is called multiscale finite volume element method (MsFVEM). We would like to note that the coarse-scalevelocity field obtained using MsFVEM is conservative in control volume elements Vξ (not in Kh).

Mixed multiscale finite element methods For simplicity, we assume Neumann boundary conditions.First, we review the mixed multiscale finite element formulation following [25] (see also [6], [1], and [7]). We

38 Thomas Y. Hou

can rewrite two-phase flow equation as

k−1u−∇p = 0 in Ω

div(u) = 0 in Ω

k(x)∇p · n = g(x) on ∂Ω.

(4.36)

The variational problem associated with (4.36) is to seek (u, p) ∈ H(div,Ω) × L2(Ω)/R such that u · n = gon ∂Ω and

(k−1u, v) + (divv, p) = 0 ∀v ∈ H0(div,Ω)

(divu, q) = 0 ∀q ∈ L2(Ω)/R.(4.37)

where H0(div,Ω) is H(div,Ω) with homogeneous Neumann boundary conditions. By defining

a(u, v) = (k−1u, v), b(v, q) = (divv, q), (4.38)

we can rewrite the weak formulation as

a(u, v) + b(v, p) = 0 ∀v ∈ H0(div,Ω),

b(u, q) = 0 ∀q ∈ L2(Ω)/R.

Let Vh ⊂ H(div,Ω) and Qh ⊂ L2(Ω)/R be finite dimensional spaces and V 0h = Vh ∩ H0(div,Ω). The

numerical approximation problem associated with (4.37) is to find (uh, ph) ∈ Vh ×Qh such that uh · n = ghon ∂Ω, where gh = g0,hn on ∂Ω and g0,h =

∑

e∈∂KT

∂Ω,K∈τh(∫

e gds)Ne, Ne ∈ Vh, is corresponding basisfunction to edge e,

(k−1uh, vh) + (divvh, ph) = 0 ∀vh ∈ V 0h

(divuh, qh) = 0 ∀qh ∈ Qh.(4.39)

One can define a linear operator Bh : V 0h → Q′

h by b(uh, qh) = (Bhuh, qh).Suppose that the following conditions are satisfied

a(uh, uh) is kerBh − coercive (4.40)

infqh∈Qh

supvh∈Vh

b(vh, qh)

‖vh‖H(div,Ω)‖qh‖L2(Ω)≥ C. (4.41)

Then the following approximation property follows (see e.g., [19]).

Lemma 41 If (u, p) and (uh, ph) respectively solve the problem (4.37) and (4.39) and the conditions (4.40)and (4.41) hold, then

‖u− uh‖H(div,Ω) + ‖p− ph‖0,Ω ≤ infvh∈Vhvh−g0,h∈V 0

h

‖u− vh‖H(div,Ω) + infqh∈Qh

‖p− qh‖0,Ω. (4.42)

Following Chen and Hou [25] (see also [6]), one can construct multiscale basis functions for velocity ineach coarse block K

div(k(x)∇wKi ) =1

|K| in K

k(x)∇wKi nK =

gKi on eKi0 else,

(4.43)

where gKi = 1|eKi| and eKi are the edges of K. Then, we can define the finite dimensional space for velocity by

Vh =⊕

K

ΨKi ,

V 0h = Vh ∩H0(div,Ω),

where ΨKi = k(x)∇wKi .


4.8 MsFEM using limited global information

Motivation Multiscale finite element methods and their modifications are used in two-phase flow simulationsthrough heterogeneous porous media. First, we briefly describe the underlying fine-scale equations. Wepresent two-phase flow equations neglecting the effects of gravity, compressibility, capillary pressure anddispersion on the fine scale. Porosity, defined as the volume fraction of the void space, will be taken tobe constant and therefore serves only to rescale time. The two phases will be referred to as water and oiland designated by the subscripts w and o, respectively. We can then write Darcy’s law, with all quantitiesdimensionless, for each phase j as follows:

vj = −λj(S)k∇p, (4.44)

where vj is phase velocity, S is water saturation (volume fraction), p is pressure, λj = krj(S)/µj is phasemobility, where krj and µj are the relative permeability and viscosity of phase j respectively, and k is thepermeability tensor, which is here taken to be diagonal, k = kI , where I is the identity matrix,

Combining Darcy’s law with conservation of mass, div(vw + vo)=0, allows us to write the flow equationin the following form

div(λ(S)k∇p) = f, (4.45)

where the total mobility λ(S) is given by λ(S) = λw(S) + λo(S) and f is a source term. The saturationdynamics affects the flow equations. One can derive the equation describing the dynamics of the saturation

∂S

∂t+ div(F) = 0, (4.46)

where F = vfw(S), with fw(S), the fractional flow of water, given by fw = λw/(λw + λo), and the totalvelocity v by:

v = vw + vo = −λ(S)k∇p. (4.47)

In the presence of capillary effects, an additional diffusion term is present in (4.46).If krw = S, kro = 1 − S and µw = µo, then the flow equation reduces to

div(k∇psp) = f.

This equation, the linear advection pollutant transport equation, will be referred to as the single-phase flowequation associated with (4.45), and psp will be referred to as the single-phase flow solution.

As we see from (4.45) and (4.46), the pressure equation is solved many times for different saturationprofiles. Thus, computing the basis functions once at time zero is very beneficial and the basis functions areonly updated near sharp interfaces. In fact, our numerical results show that only slight improvement can beachieved by updating the basis functions near sharp fronts. However, we have found that for heterogeneouspermeability fields with very strong non-local effects, the use of some type of global information can improvemultiscale finite element results significantly, which will be discussed next.

We present a representative numerical example for a permeability field generated using two-point geo-statistics. To generate this permeability field, we have used GSLIB algorithm [30]. The permeability islog-normally distributed with prescribed variance σ2 = 1.5 (σ2 here refers to the variance of log k) and somecorrelation structure. The correlation structure is specified in terms of dimensionless correlation lengths inthe x and z-directions, lx = 0.4 and lz = 0.04, nondimensionalized by the system length. Linear boundaryconditions are used for constructing multiscale basis function in (4.34). Spherical variogram is used [30]. Inthis numerical example, the fine-scale field is 120× 120, while the coarse-scale field is 12× 12 defined in therectangle with the length 5 and the width 1. For the two-phase flow simulations, the system is consideredto initially contain only oil (S = 0) and water is injected at inflow boundaries (S = 1 is prescribed), i.e., wespecify p = 1, S = 1 along the x = 0 edge and p = 0 along the x = 5 edge, and no flow boundary conditionson the lateral boundaries. Relative permeability functions are specified as krw = S2, kro = (1 − S)2; waterand oil viscosities are set to µw = 1 and µo = 5. Porosity is constant and serves only to nondimensionalizetime. Results are presented in terms of the fraction of oil in the produced fluid (i.e., oil cut, designated F )

40 Thomas Y. Hou

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

PVI

F

fineprimitivestandard MsFVEM

Fig. 4.11. Fractional flow comparison for a permeability field generated using two-point geostatistics.

fine−scale saturation plot at PVI=0.5

0

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0.5

0.6

0.7

0.8

0.9

1saturation plot at PVI=0.5 using standard MsFVEM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 4.12. Saturation maps at PVI=0.5 for fine-scale solution (left figure) and standard MsFVEM (right figure)

against pore volume injected (PVI). PVI represents dimensionless time and is computed via∫

Qdt/Vp whereVp is the total pore volume of the system and Q is the total flow rate.

In our first numerical test, Figure 4.11, we compare the fractional flows. The dashed line correspondsto the calculations performed using a simple saturation upscaling (no subgrid treatment), while dotted linecorresponds to the calculations performed by solving the saturation equation on the fine grid using thereconstructed fine-scale velocity field. We observe from this figure that the second approach is very accurate,while the first approach over-predicts the breakthrough time. The saturation snapshots are compared inFigure 4.12. One can observe that there is a very good agreement.

In the next set of numerical results, we consider strongly channelized permeability fields. These perme-ability fields are proposed in some recent benchmark tests, such as the SPE comparative solution project[27]. In Figure 4.13, one of the layers of this 3-D permeability field is depicted. All the layers have 220× 60fine-scale resolution, and we take the coarse grid to be 22 × 6. As it can be observed, the permeability fieldcontains a high permeability channel, where most flow will occur in our simulation. In Figure 4.14, the frac-tional flows are compared. The boundary conditions are taken to be p = 1, S = 1 along the x = 0 edge andp = 0 along the x = 5 edge, and no flow boundary conditions on the lateral boundaries. Again, the dashedline corresponds to the calculations performed using a simple saturation upscaling (no subgrid treatment),


−6

−4

−2

0

2

4

6

8

Fig. 4.13. Log-permeability for one of the layers of upper Ness.

while dotted line corresponds to the calculations performed by solving the saturation equation on the finegrid using the reconstructed fine-scale velocity field. We observe from this figure that the second approachis not very accurate in contrast to the permeability field generated using two-point geostatistics [30]. Thisis because the local basis functions can not account accurately the global connectivity of the media. Indeed,in the next figure, Figure 4.15, the saturation fields at time PVI = 0.5 are compared. We see that multiscalefinite element methods with local basis functions introduce some errors. In the bottom left corner, there isa saturation pocket which is not in the reference solution computed using a fine grid. The reason for this isthat the local basis functions in the lower left corner contains high permeability region. However, this highpermeability region does not have global connectivity, and the local basis functions can not take this effectinto account. Next, we discuss how global information can be incorporated into multiscale basis functions toimprove the accuracy of the computations.

Multiscale finite volume element method The main idea of the modified multiscale finite volumeelement method (MsFVEM) is to use the solution of the fine-scale problem at time zero to determine theboundary conditions for the basis functions. This approach is proposed in [40] to handle the permeabilityfields which are strongly channelized. For this type of permeability fields, some type of global information isneeded. Next, we describe the method. We denote the solution of pressure equation at time zero by psp(x).In defining psp(x), we use the actual boundary conditions of the global problem. psp(x) depends on globalboundary conditions, and, generally, is updated each time when global boundary conditions are changed.The boundary conditions in (4.34) for modified basis functions are defined in the following way. For eachrectangular element K with vertices xi (i = 1, 2, 3, 4) denote by φi(x) a restriction of the nodal basis onK, such that φi(xj) = δij . At the edges where φi(x) = 0 at both vertices, we take boundary condition forφi(x) to be zero. Consequently, the basis functions are localized. We only need to determine the boundarycondition at two edges which have the common vertex xi (φi(xi) = 1). Denote these two edges by [xi−1, xi]and [xi, xi+1] (see Figure 4.16). We only need to describe the boundary condition, gi(x), for the basis functionφi(x), along the edges [xi, xi+1] and [xi, xi−1]. If psp(xi) 6= psp(xi+1), then

gi(x)|[xi,xi+1] =psp(x) − psp(xi+1)

psp(xi) − psp(xi+1), gi(x)|[xi,xi−1] =

psp(x) − psp(xi−1)

psp(xi) − psp(xi−1).

42 Thomas Y. Hou

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

PVI

F

fineprimitivestandard MsFVEM

Fig. 4.14. Fractional flow comparison for a channelized permeability field.


0

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0.5

0.6

0.7

0.8

0.9


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 4.15. Saturation maps at PVI=0.5 for fine-scale solution (left figure) and standard MsFVEM (right figure).

If psp(xi) = psp(xi+1) 6= 0, then

gi(x)|[xi,xi+1] = φ0i (x) +

1

2psp(xi)(psp(x) − psp(xi+1)),

where φ0i (x) is a linear function on [xi, xi+1] such that φ0

i (xi) = 1 and φ0i (xi+1) = 0. Similarly,

gi+1(x)|[xi,xi+1] = φ0i+1(x) +

1

2psp(xi+1)(psp(x) − psp(xi+1)), (4.48)

where φ0i+1(x) is a linear function on [xi, xi+1] such that φ0

i+1(xi+1) = 1 and φ0i+1(xi) = 0. If psp(xi) =

psp(xi+1) 6= 0, then one can also use simply linear boundary conditions. If psp(xi) = psp(xi+1) = 0 then


x

x

xi

i−1

i+1

x i−1

i+1xxi

φ (

φ (

φ ()=1 )=0

)=0

i

i

i

φ ( )=0

φ ( )=0

i

x

x

i

Fig. 4.16. Schematic description of nodal points.

linear boundary conditions are used. In the applications considered in this paper, the initial pressure isalways positive. Finally, the basis function φi(x) is constructed by solving (4.34). The choice of the boundaryconditions for the basis functions is motivated by the analysis. In particular, we would like to recover theexact fine-scale solution along each edge if the nodal values of the pressure are equal to the values of exactfine-scale pressure. This is the underlying idea for the choice of boundary conditions. Using this property andCea’s lemma one can show that the pressure obtained from the numerical solution is equal to the underlyingfine-scale pressure.

Mixed multiscale finite element methods Next, following [1], we present a mixed multiscale finiteelement method that employs single-phase flow information. Suppose that psp solves the single-phase flowequation. We set bKi = (k∇psp|eKi ) · nK and assume that bKi is uniformly bounded. Then the new basis

functions for velocity is constructed by solving the following local problems (4.43) with gKi = bKi /βKi , where

βKi =∫

eKi

k∇psp · nKds. For further analysis, we assume that βKi 6= 0. In general, if βKi = 0 one can use

standard mixed multiscale finite element basis functions. Let NKi = k(x)∇wKi and the multiscale finite

dimensional space V 0h for velocity be defined by

Vh :=⊕

K

NKi ⊂ H(div,Ω),

V 0h := Vh ∩H0(div,Ω).

First, we will show that the resulting multiscale finite element solution for velocity is exact for single-phase flow (i.e., λ(x) = 1). Let vh|K = βKi N

Ki , then βKi is the interpolation value of the fine scale solution.

Furthermore, a direct calculation yields (vh|eKi

) · nK = k∇psp · nK . Because

divvh = βKi divNKi =

1

|K|

∫

∂K

k∇psp · nKds =1

|K|

∫

K

div(k∇pspd) = 0,

the following equation is obtained immediately

divvh = 0 in K (4.49)

vh · nK = k∇psp · nK on ∂K (4.50)

44 Thomas Y. Hou

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PVI

F

finemodified MsFVEMstandard MsFVEM

Fig. 4.17. Fractional flow comparison for standard MsFVEM and modified MsFVEM for side-to-side flow.

Because div(k∇psp) = 0, we get vh = k∇psp and the following proposition.Proposition Let βKi =

∫

eKi

k∇psp · nKds, then on each coarse block K

k∇psp = βKi NKi . (4.51)

Lemma 42 If |βKi | ≥ Ch with C is independent of h, then(1) a(uh, uh) is kerBh-coercive;

(2) infqh∈Qh supvh∈V 0h

b(vh,qh)‖vh‖H(div,Ω)‖qh‖L2(Ω)

≥ C.

Numerical results Next, we show the numerical results obtained using modified multiscale finite elementtype methods for the permeability layer depicted in Figure 4.13 and two-phase flow parameters presentedearlier. We consider two types of boundary conditions in a rectangular region [0, 5]× [0, 1]. For the first typeof boundary conditions, we specify p = 1, S = 1 along the x = 0 edge and p = 0 along the x = 5 edge. On therest of the boundaries, we assume no flow boundary condition. We call this type of the boundary conditionas side-to-side. The other type of boundary conditions is obtained by specifying p = 1, S = 1 along the x = 0edge for 0.5 ≤ z ≤ 1 and p = 0 along the x = 5 edge for 0 ≤ z ≤ 0.5. On the rest of the boundaries, weassume no flow boundary condition.

In Figure 4.17, the fractional flows are plotted for standard and modified MsFVEM. We observe fromthis figure that modified MsFVEM is more accurate and provides nearly the same fractional flow responseas the direct fine-scale calculations. In Figure 4.18, we compare the saturation fields at PVI=0.5. As we see,the saturation field obtained using modified MsFVEM is very accurate and there is no longer the saturationpocket at the left bottom corner. Thus, the modified MsFVEM captures the connectivity of the mediaaccurately.

In the next set of numerical results, we test the modified multiscale finite element methods for a differentlayer (layer 40) of SPE comparative solution project. In Figure 4.19 and 4.20, the fractional flows and totalflow rates (Q) are compared for two different boundary conditions. One can see clearly that the modifiedMsFVEM method gives nearly exact results for these integrated responses. The standard MsFVEM tendsto over-predict the total flow rate at time zero. This initial error persists at later times. This phenomena isoften observed in upscaling of two-phase flows. More numerical results and discussions can be found in [40].These numerical results demonstrate that modified multiscale finite element methods which use a limited



0

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0.9

1saturation plot at PVI=0.5 using modified MsFVEM

0

0.1

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0.3

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0.5

0.6

0.7

0.8

0.9

1

Fig. 4.18. Saturation maps at PVI=0.5 for fine-scale solution (left figure) and modified MsFVEM (right figure).Side-to-side boundary condition is used.

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

PVI

F


0 0.5 1 1.5 2

100

150

200

250

300

350

400

PVI

Q

finemodified MsFVEMstandard MsFVM

Fig. 4.19. Fractional flow (left figure) and total production (right figure) comparison for standard MsFVEM andmodified MsFVEM for side-to-side flow (layer 40).

global information are more accurate. Moreover, modified multiscale finite element methods are capable ofcapturing the long-range flow features accurately for channelized permeability fields.

For the next set of results, we consider another layer of the upper Ness (layer 59). In Figure 4.21, bothfractional flow (left figure) and total flow (right figure) are plotted. We observe that the modified MsFVEMgives almost the exact results for these quantities, while the standard MsFVEM overpredicts the total flowrate, and there are deviations in the fractional flow curve around PV I ≈ 0.6. Note that unlike the previouscase, fractional flow for standard MsFVEM is nearly exact at later times (PV I ≈ 2). In Figure 4.22,the saturation maps are plotted at PV I = 0.5. The left figure represents the fine-scale, the middle figure

46 Thomas Y. Hou

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

PVI

F


0 0.5 1 1.5 2

100

150

200

250

300

350

400

PVIQ

fine

modified MsFVEM

standard MsFVM

Fig. 4.20. Fractional flow (left figure) and total production (right figure) comparison for standard MsFVEM andmodified MsFVEM for corner-to-corner (layer 40).

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PVI

F


0 0.5 1 1.5 2

200

400

600

800

1000

PVI

Q


Fig. 4.21. Fractional flow (left figure) and total production (right figure) comparison for standard MsFVEM andmodified MsFVEM for corner-to-corner flow.

represents the results obtained using standard MsFVEM, and the right figure represents the results obtainedusing the modified MsFVEM. We observe from this figure that the saturation map obtained using standardMsFVEM has some errors. These errors are more evident near the lower left corner. The results of thesaturation map obtained using the modified MsFVEM is nearly the same as the fine-scale saturation field.It is evident from these figures that the modified MsFVEM performs better than the standard MsFVEM.

Analysis

Galerkin finite element methods with limited global information We have proposed some analysisfor modified multiscale finite element method in [40] and [2]. The main idea is to show that the pressureevolution in two-phase flow simulations is strongly influenced by the initial pressure. To demonstrate this,we consider a channelized permeability field, where the value of the permeability in the channel is large.



0

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0

0.1

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0.5

0.6

0.7

0.8

0.9

1saturation plot at PVI=0.5 using modified MsFVEM

0

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0.5

0.6

0.7

0.8

0.9

1

Fig. 4.22. Saturation maps at PVI=0.5 for fine-scale solution (left figure), standard MsFVEM (middle figure), andmodified MsFVEM (right figure). Corner-to-corner boundary condition is used.

We assume the permeability has the form kI , where I is an identity matrix. In a channelized medium, thedominant flow is within the channels. Our analysis assumes a single channel and restricted to 2-D. Here, webriefly mention the main findings. Denote the initial stream function and pressure by η = ψ(x, t = 0) andζ = p(x, t = 0) (ζ is also denoted by psp previously). The stream function is defined

∂ψ/∂x1 = −v2, ∂ψ/∂x2 = v1. (4.52)

Then the equation for the pressure can be written as

∂

∂η

(

|k|2λ(S)∂p

∂η

)

+∂

∂ζ

(

λ(S)∂p

∂ζ

)

= 0. (4.53)

For simplicity, S = 0 at time zero is assumed. We consider a typical boundary condition that gives high flowwithin the channel, such that the high flow channel will be mapped into a large slab in (η, ζ) coordinatesystem. If the heterogeneities within the channel in η direction is not strong (e.g., narrow channel in Cartesiancoordinates), the saturation within the channel will depend on ζ. In this case, the leading order pressure willdepend only on ζ, and it can be shown that

p(η, ζ, t) = p0(ζ, t) + high order terms, (4.54)

where p0(ζ, t) is the dominant pressure. This asymptotic expansion shows that the time-varying pressurestrongly depends on the initial pressure (i.e., the leading order term in the asymptotic expansion is a functionof initial pressure and time only). In our analysis, we will assume that |p(x, t) − p(psp, t)|H1 is small.

Since the analysis of the multiscale finite element methods is carried out only for the pressure equation, wewill assume t (time) is fixed. Then, assuming the function p is sufficiently smooth, one can state the following.There existsAK in eachK, such that ‖∇p(x)−AK∇psp(x)‖L2(Ω) is small. Note that this assumption indicatesthat the fine-scale features of pressure solutions of two-phase equations does not change significantly during asimulation (e.g., streamlines do not vary significantly in each coarse block). This phenomena can be observedin numerical simulations of two-phase flows when µo/µw > 1.

The assumption for the case with scale separation indicates that the coarse-scale features of two-phaseflow and single-phase flow are similar (e.g., coarse-scale streamlines do not vary significantly). We will usethe following assumption.

Assumption G. There exists a sufficiently smooth scalar valued function G(η) (G ∈ C3), such that

|p−G(psp)|1,Ω ≤ Cδ, (4.55)

where psp is single-phase flow pressure and δ is sufficiently small.We note G is p0(ζ, t) at fixed t in (4.54). Moreover, one does not need to know the function G for

computing the multiscale approximation of the solution. It is only necessary that G has certain smoothnessproperties, however, it is important that the basis functions span psp in each coarse block.

48 Thomas Y. Hou

Theorem 43 Under Assumption G and psp ∈ W 1,s(Ω) (s > 2), multiscale finite element method convergeswith the rate given by

|p− ph|1,Ω ≤ Cδ + Ch1−2/s|psp|W 1,s(Ω) + Ch1−2/s|psp|1,Ω + Ch‖f‖0,Ω ≤ Cδ + Ch1−2/s. (4.56)

The proof of this theorem is given in [2]. Note that Theorem(43) shows that MsFEM converges forproblems without any scale separation and the proof of this theorem does not use homogenization techniques.Next, we present the proof.

Proof. Following standard practice of finite element estimation, we seek qh = ciφKi , where φKi are single-phase

flow based multiscale finite element basis functions. Then from Cea’s lemma, we have

|p− ph|1,Ω ≤ |p−G(psp)|1,Ω + |G(psp) − ciφKi |1,Ω. (4.57)

Next, we present an estimate for the second term. We choose ci = G(psp(xi)), where xi are vertices of K.Furthermore, using Taylor expansion of G around pK , which is the average of psp over K,

G(psp(xi)) = G(pK) +G′(pK)(psp(xi) − pK) +1

2G′′(ξxi)(p

sp(xi) − pK)2,

where ξxi = pK + θ(psp(xi) − pK), 0 < θ < 1, we have in each K

ciφKi = G(pK)φKi +G′(pK)(psp(xi) − pK)φKi +

1

2G′′(ξxi)(p

sp(xi) − pK)2φKi =

G(pK) +G′(pK)(psp(xi)φKi − pK) +

1

2G′′(ξxi)(p

sp(xi) − pK)2φKi .

(4.58)

In the last step, we have used∑

i φKi = 1. Similarly, in each K,

G(psp(x)) = G(pK) +G′(pK)(psp(x) − pK) +1

2G′′(ξx)(p

sp(x) − pK)2, (4.59)

where ξx = pK + θ(psp(x) − pK), 0 < θ < 1. Using (4.58) and (4.59), we get

|G(psp) − ciφKi |1,K ≤ |G′(pK)(psp(x) − psp(xi)φ

Ki )|1,K + |1

2G′′(ξxi)(p

sp(xi) − pK)2φKi |1,K+

|12G′′(ξx)(p

sp(x) − pK)2|1,K .(4.60)

Because of |psp(x) − psp(xi)|1,K ≤ Ch‖f‖0,K , the estimate of the first term is the following

|G′(pK)(psp(x) − psp(xi)φKi )|1,K ≤ Ch‖f‖0,K.

For the second term on the right hand side of (4.60), assuming psp(x) ∈W 1,s(Ω), we have

|G′′(ξxi)(psp(xi) − pK)2φKi |1,K ≤ Ch2−4/s|psp|2W 1,s(K) ≤ Ch1−2/s|psp|W 1,s(K).

where s > 2. Here, we have used the inequality (e.g., [4])

|u(x) − u(y)| ≤ C|x − y|1−2/s|u|W 1,s ,

for s > 2, where C depends only on s.For the third term, since G′′ and G′′′ are bounded, we have the following estimate:

|G′′(ξx)(psp(x) − pK)2|1,K ≤ C‖(psp(x) − pK)2∇psp(x)‖0,K+

C‖(psp(x) − pK)∇psp(x)‖0,K ≤ Ch2−4/s|psp|2W 1,s(K)|psp|1,K+

Ch1−2/s|psp|1,K ≤ Ch2−4/s|psp|2W 1,s(Ω)|psp|1,K+

Ch1−2/s|psp|1,K ≤ Ch1−2/s|psp|1,K .

(4.61)


Combining the above estimates, we have for (4.60),

|G(psp) − ciφKi |1,K ≤ Ch1−2/s|psp|W 1,s(K) + Ch1−2/s|psp|1,K + Ch‖f‖0,K . (4.62)

Summing (4.62) over all K and taking into account Assumption G, we have

|p− ph|1,Ω ≤ Cδ + Ch1−2/s|psp|W 1,s(Ω) + Ch1−2/s|psp|1,Ω + Ch‖f‖0,Ω ≤ Cδ + Ch1−2/s. (4.63)

Consequently, if s > 2 (see e.g., [9]) single-phase flow based multiscale finite element method converges.

Extensions of Galerkin finite element methods with limited global information The multiscalefinite element methods considered above employ information from only one single-phase flow solution. Ingeneral, depending on the source term, boundary data, and mobility λ(S) (if it contains sharp variations),it might be necessary to use information from multiple global solutions for the computation of accurate two-phase flow solution. The previous multiscale finite element methods can be extended to take into accountadditional global information. Next, we present an extension of the Galerkin multiscale finite element methodthat uses the partition of unity method [12] (also see e.g., [96], [56], [69]).

Assume that u1, u2,..., uN are the global functions such that |p−G(u1, u2, ..., uN )|1,Ω is sufficiently small.Here, u1, ..., uN can be possible pressure snapshots for different mobility λ(S) or pressure fields correspondingto different source terms and/or boundary conditions. We would like to note that in a very interesting paper[88], the authors prove under certain conditions on f (source term) and λ = 1 that p is a smooth functionof single-phase flow solutions (elliptic pressure equations) with boundary conditions x1 and x2 (it is alsoextended to multi-dimensional case). In this case, u1 and u2 are the solutions of single-phase flow equationswith boundary conditions ui = xi (i = 1, 2), and it was shown that p(u1, u2) ∈ H2. Next, we will formulatethe method.

Let ωi be a patch (see Figure 4.23), and define φ0i to be piecewise linear basis function in patch ωi, such

that φ0i (xj) = δij . For simplicity of notation, denote u1 = 1. Then, the multiscale finite element method for

each patch ωi is constructed byψij = φ0

i uj (4.64)

where j = 1, .., N and i is the index of nodes (see Figure 4.23). First, we note that in each K,∑n

i=1 ψij = ujis the desired single-phase flow solution.

i

xi

ωK

Fig. 4.23. Schematic description of patch

We will use the following assumption. There exists a sufficiently smooth scalar valued function G(η),η ∈ RN (G ∈ C3), such that

|p−G(u1, ..., uN )|1,Ω ≤ Cδ, (4.65)

where δ is sufficiently small.

50 Thomas Y. Hou

As before the form of the function G is not important for the computations, however, it is crucial thatthe basis functions span u1,..., uN in each coarse block. The next theorem shows that MsFEM converges forproblems without scale separation in this case.

Theorem 44 Assume (4.65) and ui ∈W 1,s(Ω), s > 2, i = 1, ..., N . Then

|p− ph|1,Ω ≤ Cδ + Ch1−2/s.

The proof of this theorem is given in [2].

Mixed finite element methods with limited global information One can carry out the analysis ofmixed multiscale finite element method with limited global information. First, we re-formulate our assump-tion for the analysis of mixed multiscale finite element methods. From (4.55), it follows that

‖∇p−G′(psp)∇psp‖0,Ω ≤ Cδ.

Using the fact that k and λ(x) are bounded, we have

‖λ(x)k∇p−G′(psp)λ(x)k∇psp‖0,Ω ≤ Cδ.

Noting that u = λ(x)k∇p and usp = k∇psp, it follows that there exists a coarse-scale function scalar A(x)such that

‖u−A(x)usp‖0,Ω ≤ δ. (4.66)

Since A(x)usp approximates u, we assume that it has small divergence,

|∫

K

div(A(x)usp)dx| ≤ Cδ1h2 (4.67)

in each K, where δ1 is a small number. For our analysis, we note that (4.67) gives

|∫

∂K

A(x)uspnKds| ≤ Cδ1h2. (4.68)

We will assume that A(x) ∈ Cγ (0 < γ ≤ 1). (4.68) can be written as

|∑

i

Ai

∫

∂eKi

uspnKds| ≤ Cδ1h2. (4.69)

Here Ai’s are defined as Ai =∫

∂eKi

A(x)uspnKds/∫

∂eKi

uspnKds, since∫

∂eKi

uspnKds = βKi 6= 0. Note that

not for any A(x), Ai is necessarily a value of A(x) along the edge eKi because uspnK can change sign.However, we only need to define A(x) for each edge by its value Ai (e.g., the value of A(x) at the center ofedge). Then, for any such A(x), (4.66) is satisfied provided δ < hγ . This can be directly verified. Thus, ourmain assumption will be (4.66) and (4.69), where A(x) is defined, for example, at the center of each edgeeKi . We would like to note that from the fact that div(A(x)usp) is small in each K, it follows that A(x), forexample, can be taken as an approximation of stream function corresponding to usp. As before, the form ofA(x) is not important for the computations of multiscale solutions.

The following theorem about the convergence of mixed multiscale finite element methods for problemswithout scale separation is proven in [2].

Theorem 45 Assume (4.66) and (4.69) and A(x) ∈ Cγ , 0 < γ ≤ 1. Let (u, p) and (uh, ph) respectively solvethe problem (4.37) and (4.39) with single-phase flow based mixed multiscale finite element, then

‖u− uh‖H(div,Ω) + ‖p− ph‖0,Ω ≤ Cδ + Cδ1 + Chγ . (4.70)


5 Multiscale finite element methods for nonlinear partial differential equations

Next, we show that MsFEM can be naturally generalized to nonlinear partial differential equations. Thegoal of MsFEM is to find a numerical approximation of a homogenized solution without solving auxiliaryproblems (e.g., periodic cell problems) that arise in homogenization. The homogenized solutions are soughton a coarse grid space Sh, where h ε. Let Kh be a partition of Ω. We denote by Sh standard familyof finite dimensional space, which possesses approximation properties, e.g., piecewise linear functions overtriangular elements,

Sh = vh ∈ C0(Ω) : the restriction vh is linear for each element K and vh = 0 on ∂Ω. (5.1)

In further presentation, K is a triangular element that belongs to Kh. To formulate MsFEM for generalnonlinear problems, we will need (1) a multiscale mapping that gives us the desired approximation containingthe small scale information and (2) a multiscale numerical formulation of the equation.

We consider the formulation and analysis of MsFEM for general nonlinear elliptic equations, uε ∈W 1,p0 (Ω)

−divaε(x, uε,∇uε) + a0,ε(x, uε,∇uε) = f, (5.2)

where aε(x, η, ξ) and a0,ε(x, η, ξ), η ∈ R, ξ ∈ Rd satisfy the following assumptions:

|aε(x, η, ξ)| + |a0,ε(x, η, ξ)| ≤ C (1 + |η|p−1 + |ξ|p−1), (5.3)

(aε(x, η, ξ1) − aε(x, η, ξ2), ξ1 − ξ2) ≥ C |ξ1 − ξ2|p, (5.4)

(aε(x, η, ξ), ξ) + a0,ε(x, η, ξ)η ≥ C|ξ|p. (5.5)

Denote

H(η1, ξ1, η2, ξ2, r) = (1 + |η1|r + |η2|r + |ξ1|r + |ξ2|r), (5.6)

for arbitrary η1, η2 ∈ R, ξ1, ξ2 ∈ Rd, and r > 0. We further assume that

|aε(x, η1, ξ1) − aε(x, η2, ξ2)| + |a0,ε(x, η1, ξ1) − a0,ε(x, η2, ξ2)|≤ C H(η1, ξ1, η2, ξ2, p− 1) ν(|η1 − η2|)+ C H(η1, ξ1, η2, ξ2, p− 1 − s) |ξ1 − ξ2|s,

(5.7)

where η ∈ R and ξ ∈ Rd, s > 0, p > 1, s ∈ (0,min(p− 1, 1)) and ν is the modulus of continuity, a bounded,

concave, and continuous function in R+ such that ν(0) = 0, ν(t) = 1 for t ≥ 1 and ν(t) > 0 for t > 0. Theseassumptions guarantee the well-posedness of the nonlinear elliptic problem (5.2). Here Ω ⊂ R

d is a Lipschitzdomain and ε denotes the small scale of the problem. The homogenization of nonlinear partial differentialequations has been studied previously (see, e.g., [89]). It can be shown that a solution uε converges (up to asub-sequence) to u in an appropriate norm, where u ∈ W 1,p

0 (Ω) is a solution of a homogenized equation

−diva∗(x, u,Du) + a∗0(x, u,Du) = f. (5.8)

Multiscale mapping. Introduce the mapping EMsFEM : Sh → V hε in the following way. For each elementvh ∈ Sh, vε,h = EMsFEMvh is defined as the solution of

−divaε(x, ηvh ,∇vε,h) = 0 in K, (5.9)

vε,h = vh on ∂K and ηvh = 1|K|∫

Kvhdx for each K. We would like to point out that different boundary

conditions can be chosen to obtain more accurate solutions and this will be discussed later. Note that forlinear problems, EMsFEM is a linear operator, where for each vh ∈ Sh, vε,h is the solution of the linearproblem. Consequently, V hε is a linear space that can be obtained by mapping a basis of Sh. This is preciselythe construction presented in [62] for linear elliptic equations.

52 Thomas Y. Hou

Multiscale numerical formulation. Multiscale finite element formulation of the problem is the following.Find uh ∈ Sh (consequently, uε,h(= EMsFEMuh) ∈ V hε ) such that

〈Aε,huh, vh〉 =

∫

Ω

fvhdx ∀vh ∈ Sh, (5.10)

where

〈Aε,huh, vh〉 =∑

K∈Kh

∫

K

((aε(x, ηuh ,∇uε,h),∇vh) + a0,ε(x, η

uh ,∇uε,h)vh)dx. (5.11)

Note that the above formulation of MsFEM is a generalization of the Petrov-Galerkin MsFEM introducedin [60] for linear problems. MsFEM, introduced above, can be generalized to different kinds of nonlinearproblems and this will be discussed later.

5.1 Multiscale finite volume element method (MsFVEM)

The formulation of multiscale finite element (MsFEM) can be extended to a finite volume method. Byits construction, the finite volume method has local conservative properties [53] and it is derived from alocal relation, namely the balance equation/conservation expression on a number of subdomains which arecalled control volumes. Finite volume element method can be considered as a Petrov-Galerkin finite elementmethod, where the test functions are constants defined in a dual grid. Consider a triangle K, and let zKbe its barycenter. The triangle K is divided into three quadrilaterals of equal area by connecting zK to themidpoints of its three edges. We denote these quadrilaterals by Kz, where z ∈ Zh(K) are the vertices ofK. Also we denote Zh =

⋃

K Zh(K), and Z0h are all vertices that do not lie on ΓD, where ΓD is Dirichlet

boundaries. The control volume Vz is defined as the union of the quadrilaterals Kz sharing the vertex z(see Figure 5.1). The multiscale finite volume element method (MsFVEM) is to find uh ∈ Sh (consequently,

K

Vz

z

K

zK

z

Kz

Fig. 5.1. Left: Portion of triangulation sharing a common vertex z and its control volume. Right: Partition of atriangle K into three quadrilaterals

uε,h = EMsFV EMuh such that

−∫

∂Vz

aε (x, ηuh ,∇uε,h) · n dS +

∫

Vz

a0,ε (x, ηuh ,∇uε,h) dx =

∫

Vz

f dx ∀z ∈ Z0h, (5.12)

where n is the unit normal vector pointing outward on ∂Vz . Note that the number of control volumes thatsatisfies (5.12) is the same as the dimension of Sh. We will present numerical results for both multiscalefinite element and multiscale finite volume element methods.


5.2 Examples of V h

ε

Linear case. For linear operators, V hε can be obtained by mapping a basis of Sh. Define a basis of Sh,Sh = span(φi0), where φi0 are standard linear basis functions. In each element K ∈ Kh, we define a set ofnodal basis φiε, i = 1, . . . , nd with nd(= 3) being the number of nodes of the element, satisfying

−divaε(x)∇φiε = 0 in K ∈ Kh (5.13)

and φiε = φi0 on ∂K. Thus, we have

V hε = spanφiε; i = 1, . . . , nd, K ⊂ Kh ⊂ H10 (Ω).

Oversampling technique can be used to improve the method [62].Special nonlinear case. For the special case, aε(x, uε,∇uε) = aε(x)b(uε)∇uε, V hε can be related to the

linear case. Indeed, for this case, the local problems associated with the multiscale mapping EMsFEM (see(5.9)) have the form

−divaε(x)b(ηvh )∇vε,h = 0 in K.

Because ηvh are constants over K, the local problems satisfy the linear equations,

−divaε(x)∇φiε = 0 in K,

and V hε can be obtained by mapping a basis of Sh as it is done for the first example. Thus, for this case onecan construct the base functions in the beginning of the computations.

V hε using subdomain problems. One can use the solutions of smaller (than K ∈ Kh) subdomainproblems to approximate the solutions of the local problems (5.9). This can be done in various ways basedon a homogenization expansion. For example, instead of solving (5.9) we can solve (5.9) in a subdomainS with boundary conditions vh restricted onto the subdomain boundaries, ∂S. Then the gradient of thesolution in a subdomain can be extended periodically to K to approximate ∇vε,h in (5.11). vε,h can beeasily reconstructed based on ∇vε,h. When the multiscale coefficient has a periodic structure, the multiscalemapping can be constructed over one periodic cell with a specified average.

5.3 Convergence of MsFEM for nonlinear partial differential equations

In [44] it was shown using G-convergence theory that

limh→0

limε→0

‖uh − u‖W 1,p0 (Ω) = 0, (5.14)

(up to a subsequence) where u is a solution of (5.8) and uh is a MsFEM solution given by (5.10). Thisresult can be obtained without any assumption on the nature of the heterogeneities and can not be improvedbecause there could be infinitely many scales, α(ε), present such that α(ε) → 0 as ε→ 0.

For the periodic case, it can be shown that the convergence of MsFEM in the limit as ε/h→ 0. To showthe convergence for ε/h→ 0, we consider h = h(ε), such that h(ε) ε and h(ε) → 0 as ε→ 0. We would liketo note that this limit as well as the proof of the periodic case is different from (5.14), where the double-limitis taken. In contrast to the proof of (5.14), the proof of the periodic case requires the correctors for thesolutions of the local problems.

Next we will present the convergence results for MsFEM solutions. For general nonlinear elliptic equationsunder the assumptions (5.3)-(5.7) the strong convergence of MsFEM solutions can be shown. In the proofof this theorem we show the form of the truncation error (in a weak sense) in terms of the resonanceerrors between the mesh size and small scale ε. The resonance errors are derived explicitly. To obtain theconvergence rate from the truncation error, one needs some lower bounds. Under the general conditions,such as (5.3)-(5.7), one can prove strong convergence of MsFEM solutions without an explicit convergencerate (cf. [94]). To convert the obtained convergence rates for the truncation errors into the convergence rateof MsFEM solutions, additional assumptions, such as monotonicity, are needed.

Next, we formulate convergence theorems. The proofs can be found in [41].

54 Thomas Y. Hou

Theorem 51 Assume aε(x, η, ξ) and a0,ε(x, η, ξ) are periodic functions with respect to x and let u be a solutionof (5.8) and uh is a MsFEM solution given by (5.10). Moreover, we assume that ∇uh is uniformly boundedin Lp+α(Ω) for some α > 0. Then

limε→0

‖uh − u‖W 1,p0 (Ω) = 0 (5.15)

where h = h(ε) ε and h→ 0 as ε→ 0 (up to a subsequence).

Theorem 52 Let u and uh be the solutions of the homogenized problem (5.8) and MsFEM (5.10), respectively,with the coefficient aε(x, η, ξ) = a(x/ε, ξ) and a0,ε = 0. Then

‖uh − u‖pW 1,p

0 (Ω)≤ c

( ε

h

)s

(p−1)(p−s)

+ c( ε

h

)

pp−1

+ chpp−1 . (5.16)

5.4 Multiscale finite element methods for nonlinear parabolic equations

We consider∂

∂tuε − div(aε(x, t, uε,∇uε)) + a0,ε(x, t, uε,∇uε) = f, (5.17)

where ε is a small scale. Our motivation in considering (5.17) mostly stems from the applications of flow inporous media (multi-phase flow in saturated porous media, flow in unsaturated porous media) though manyapplications of nonlinear parabolic equations of these kinds occur in transport problems. Many problems insubsurface modeling have multiscale nature where the heterogeneities associated with the media is no longerperiodic. It was shown that a solution uε converges to u (up to a subsequence) in an appropriate sense whereu is a solution of

∂

∂tu− div(a∗(x, t, u,∇u)) + a∗0(x, t, u,∇u) = f. (5.18)

In [43] the homogenized fluxes a∗ and a∗0 are computed under the assumption that the heterogeneities arestrictly stationary random fields with respect to both space and time.

The numerical homogenization procedure presented in the previous section can be extended to parabolicequations. To do this we will first formulate MsFEM in a slightly different manner from that presented in[62] for the linear problem. Consider a standard finite dimensional Sh space over a coarse triangulation ofΩ, (5.1) and define EMsFEM : Sh → V hε in the following way. For each uh ∈ Sh there is a correspondingelement uh,ε in V hε that is defined by

∂

∂tuh,ε − div(aε(x, t)∇uh,ε) = 0 in K × [tn, tn+1], (5.19)

with boundary condition uh,ε = uh on ∂K, and uh,ε(t = tn) = uh. For the linear equations EMsFEM is alinear operator and the obtained multiscale space, V h

ε is a linear space on Ω× [tn, tn+1]. Moreover, the basisin the space V hε can be obtained by mapping the basis functions of Sh. Forthe nonlinear parabolic equationsconsidered in this paper the operator EMsFEM is constructed similar to (5.19) using the local problems, i.e.,for each uh ∈ Sh there is a corresponding element uh,ε in V hε that is defined by

∂

∂tuh,ε − div(aε(x, t, η,∇uh,ε)) = 0 in K × [tn, tn+1], (5.20)

with boundary condition uh,ε = uh on ∂K, and uh,ε(t = tn) = uh. Here η = 1|K|∫

Kuhdx. Note EMsFEM is

a nonlinear operator and V hε is no longer a linear space.The following method that can be derived from general multiscale finite element framework is equivalent

to our numerical homogenization procedure. Find uh ∈ V hε such that

∫ tn+1

tn

∫

Ω

∂

∂tuhvhdxdt +A(uh, vh) =

∫ tn+1

tn

∫

Ω

fvhdxdt, ∀vh ∈ Sh,


where

A(uh, wh) =∑

K

∫ tn+1

tn

∫

K

(aε(x, t, ηuh ,∇vε),∇wh) + a0,ε(x, t, η

uh ,∇vε)wh)dxdt,

where vε is the solution of the local problem 5.20), uh = luh in each K, ηuh = 1|K|∫

Kluhdx, and uh is known

at t = tn.

We would like to note that the operator EMsFEM can be constructed using larger domains as it is donein MsFEM with oversampling [62]. This way one reduces the effects of the boundary conditions and initialconditions. In particular, for the temporal oversampling it is only sufficient to start the computations beforetn and end them at tn+1. Consequently, the oversampling domain for K× [tn, tn+1] consists of [tn, tn+1]×S,where tn < tn and K ⊂ S. More precise formulation and detail numerical studies of oversampling techniquefor nonlinear equations are currently under investigation. Further we would like to note that oscillatory initialconditions can be imposed (without using oversampling techniques) based on the solution of the elliptic partof the local problems (5.20). These initial conditions at t = tn are the solutions of

−div(aε(x, t, η,∇uh,ε)) = 0 in K, (5.21)

or

−div(aε(x, η,∇uh,ε)) = 0 in K, (5.22)

where aε(x, η, ξ) = 1tn+1−tn

∫ tn+1

tna(T (x/εβ, τ/εα)ω, η, ξ)dτ and uh,ε = uh on ∂K. The latter can become

efficient depending on the inter-play between the temporal and spatial scales. This issue is discussed below.

Note that in the case of periodic media the local problems can be solved in a single period in order toconstruct A(uh, vh). In general, one can solve the local problems in a domain different from K (an element)to calculate A(uh, vh), and our analysis is applicable to these cases. Note that the numerical advantages ofour approach over the fine scale simulation is similar to that of MsFEM. In particular, for each Newton’siteration a linear system of equations on a coarse grid is solved.

For some special cases the operator EMsFEM introduced in the previous section can be simplified (see[44]). In general one can avoid solving the local parabolic problems if the ratio between temporal and spa-tial scales is known, and solve instead a simplified equation. For example, assuming that aε(x, t, η, ξ) =a(x/εβ , t/εα, η, ξ), we have the following. If α < 2β one can solve instead of (5.20) the local problem−div(aε(x, t, ηuh ,∇vε)) = 0, if α > 2β one can solve instead of (5.20) the local problem −div(aε(x, ηuh ,∇vε)) =0, where aε(x, η, ξ) is an average over time of aε(x, t, η, ξ), while if α = 2β we need to solve the parabolicequation in K × [tn, tn+1], (5.20).

We would like to note that, in general, one can use (5.21) or (5.22) as oscillatory initial conditions andthese initial conditions can be efficient for some cases. For example, for α > 2β with initial conditions givenby (5.22) the solutions of the local problems (5.20) can be computed easily since they are approximated by(5.22). Moreover, one can expect better accuracy with (5.22) for the case α > 2β because this initial conditionis more compatible with the local heterogeneities compare to the artificial linear initial conditions (cf. (5.20)).The comparison of various oscillatory initial conditions including the ones obtained by oversampling methodis a subject of future studies.

Finally, we would like to mention that one can prove the following theorem.

Theorem 53 uh =∑

i θi(t)φ0i (x) converges to u, a solution of the homogenized equation in V0 = Lp(0, T,W 1,p

0 (Ω))as limh→0 limε→0 under additional not restrictive assumptions (see [44]).

Remark 5.1. The proof of the theorem uses the convergence of the solutions and the fluxes, and consequentlyit is applicable for the case of general heterogeneities that usesG-convergence theory. Since the G-convergenceof the operators occurs up to a subsequence the numerical solution converges to a solution of a homogenizedequation (up to a subsequence of ε).

56 Thomas Y. Hou

5.5 Numerical results

In this section we present several ingredients pertaining to the implementation of multiscale finite elementmethod for nonlinear elliptic equations. More numerical examples relevant to subsurface applications can befound in [41]. We will present numerical results for both MsFEM and multiscale finite volume element method(MsFVEM). We use an Inexact-Newton algorithm as an iterative technique to tackle the nonlinearity. For

the numerical examples below, we use aε(x, uε,∇uε) = aε(x, uε)∇uε. Let φi0Ndofi=1 be the standard piecewise

linear basis functions of Sh. Then MsFEM solution may be written as

uh =

Ndof∑

i=1

αi φi0 (5.23)

for some α = (α1, α2, · · · , αNdof )T , where αi depends on ε. Hence, we need to find α such that

F (α) = 0, (5.24)

where F : RNdof → R

Ndof is a nonlinear operator such that

Fi(α) =∑

K∈Kh

∫

K

(aε(x, ηuh )∇uε,h),∇φi0) dx−

∫

Ω

f φi0 dx. (5.25)

We note that in (5.25) α is implicitly buried in ηuh and uε,h. An inexact-Newton algorithm is a variationof Newton’s iteration for nonlinear system of equations, where the Jacobian system is only approximatelysolved. To be specific, given an initial iterate α0, for k = 0, 1, 2, · · · until convergence do the following:

• Solve F ′(αk)δk = −F (αk) by some iterative technique until ‖F (αk) + F ′(αk)δk‖ ≤ βk ‖F (αk)‖.• Update αk+1 = αk + δk.

In this algorithm F ′(αk) is the Jacobian matrix evaluated at iteration k. We note that when βk = 0 then wehave recovered the classical Newton iteration. Here we have used

βk = 0.001

( ‖F (αk)‖‖F (αk−1)‖

)2

, (5.26)

with β0 = 0.001. Choosing βk this way, we avoid over-solving the Jacobian system when αk is still considerablyfar from the exact solution.

Next we present the entries of the Jacobian matrix. For this purpose, we use the following notations.Let Kh

i = K ∈ Kh : zi is a vertex of K, I i = j : zj is a vertex of K ∈ Khi , and Kh

ij = K ∈ Khi :

K shares zizj. We note that we may write Fi(α) as follows:

Fi(α) =∑

K∈Khi

(∫

K

(aε(x, ηuh )∇uε,h,∇φi0) dx−

∫

K

f φi0 dx

)

, (5.27)

with

−divaε(x, ηuh )∇uε,h = 0 in K and uε,h =∑

zm∈ZKαm φ

m0 on ∂K, (5.28)

where ZK is all the vertices of elementK. It is apparent that Fi(α) is not fully dependent on all α1, α2, · · · , αd.Consequently, ∂Fi(α)

∂αj= 0 for j /∈ I i. To this end, we denote ψjε =

∂uε,h∂αj

. By applying chain rule of differenti-

ation to (5.28) we have the following local problem for ψjε :

−divaε(x, ηuh)∇ψjε =1

3div

∂aε(x, ηuh )

∂u∇uε,h in K and ψjε = φjε on ∂K. (5.29)


The fraction 1/3 comes from taking the derivative in the chain rule of differentiation. In the formulationof the local problem, we have replaced the nonlinearity in the coefficient by ηvh , where for each triangleK ηvh = 1/3

∑3i=1 α

Ki , which gives ∂ηvh/∂αi = 1/3. Moreover, for a rectangular element the fraction 1/3

should be replaced by 1/4.Thus, provided that vε,h has been computed, then we may compute ψjε using (5.29). Using the above

descriptions we have the expressions for the entries of the Jacobian matrix:

∂Fi∂αi

=∑

K∈Khi

(

1

3

∫

K

(∂aε(x, η

uh)

∂u∇uε,h,∇φi0) dx +

∫

K

(aε(x, ηuh)∇ψi,∇φi0) dx,

)

(5.30)

∂Fi∂αj

=∑

K∈Khij

(

1

3

∫

K

(∂aε(x, η

uh )

∂u∇uε,h,∇φiε) dx +

∫

K

(aε(x, ηuh)∇ψjε ,∇φi0) dx,

)

(5.31)

for j 6= i, j ∈ I i.The implementation of the oversampling technique is similar to the procedure presented earlier, except

the local problems in larger domains are used. As in the non-oversampling case, we denote ψjε =∂vε,h∂αj

, such

that after applying chain rule of differentiation to the local problem we have:

−divaε(x, ηuh )∇ψjε =1

3div

∂aε(x, ηuh )

∂u∇vε,h in S and ψjε = φj0 on ∂S, (5.32)

where ηuh is computed over the corresponding element K and φj0 is understood as the nodal basis functionson oversampled domain S. Then all the rest of the inexact-Newton algorithms are the same as in the non-oversampling case. Specifically, we also use (5.30) and (5.31) to construct the Jacobian matrix of the system.We note that we will only use ψjε from (5.32) pertaining to the element K.

From the derivation (both for oversampling and non-oversampling) it is obvious that the Jacobian matrixis not symmetric but sparse. Computation of this Jacobian matrix is similar to computing the stiffness matrixresulting from standard finite element, where each entry is formed by accumulation of element by elementcontribution. Once we have the matrix stored in memory, then its action to a vector is straightforward.Because it is a sparse matrix, devoting some amount of memory for entries storage is inexpensive. Theresulting linear system is solved using preconditioned bi-conjugate gradient stabilized method.

We want to solve the following problem:

−diva(x/ε, uε)∇uε = −1 in Ω ⊂ R2,

uε = 0 on ∂Ω,(5.33)

where Ω = [0, 1]× [0, 1], a(x/ε, uε) = k(x/ε)/ (1 + uε)l(x/ε)

, with

k(x/ε) =2 + 1.8 sin(2πx1/ε)

2 + 1.8 cos(2πx2/ε)+

2 + sin(2πx2/ε)

2 + 1.8 cos(2πx1/ε)(5.34)

and l(x/ε) is generated from k(x/ε) such that the average of l(x/ε) over Ω is 2. Here we use ε = 0.01.Because the exact solution for this problem is not available, we use a well resolved numerical solutionusing standard finite element method as a reference solution. The resulting nonlinear system is solved usinginexact-Newton algorithm. The reference solution is solved on 512 × 512 mesh. Tables 5.2 and 5.4 presentthe relative errors of the solution with and without oversampling, respectively. In tables 5.3 and 5.5, therelative errors for multiscale finite volume element method are presented. The relative errors are computedas the corresponding error divided by the norm of the solution. In each table, the second, third, and fourthcolumns list the relative error in L2, H1, and L∞ norm, respectively. As we can see from these two tables,the oversampling significantly improves the accuracy of the multiscale method.

For our next example, we consider the problem with non-periodic coefficients, where aε(x, η) = kε(x)/(1+η)αε(x). kε(x) = exp(βε(x)) is chosen such that βε(x) is a realization of a random field with the spherical

58 Thomas Y. Hou

variogram [30] and with the correlation lengths lx = 0.2, ly = 0.02 and with the variance σ = 1. αε(x) ischosen such that αε(x) = kε(x) + const with the spatial average of 2. As for the boundary conditions we use“left-to-right flow” in Ω = [0, 5]×[0, 1] domain, uε = 1 at the inlet (x1 = 0), uε = 0 at the outlet (x1 = 5), andno flow boundary conditions on the lateral sides x2 = 0 and x2 = 1. In Table 5.6 we present the relative errorfor multiscale method with oversampling. Similarly, in Table 5.7 we present the relative error for multiscalefinite volume method with oversampling. Clearly, the oversampling method captures the effects induced bythe large correlation features. Both H1 and horizontal flux errors are under five percent. Similar resultshave been observed for various kinds of non-periodic heterogeneities. In the next set of numerical examples,we test MsFEM for problems with fluxes aε(x, η) that are discontinuous in space. The discontinuity in thefluxes is introduced by multiplying the underlying permeability function, kε(x), by a constant in certainregions, while leaving it unchanged in the rest of the domain. As an underlying permeability field, kε(x), wechoose the random field used for the results in Table 5.6. In the first set of examples, the discontinuities areintroduced along the boundaries of the coarse elements. In particular, kε(x) on the left half of the domain ismultiplied by a constant J , where J = exp(1), or exp(2), or exp(4). The results in Tables 5.8-5.10 show thatMsFEM converges and the error falls below five percent for relatively large coarsening. For the second set ofexamples (Tables 5.11-5.13), the discontinuities are not aligned with the boundaries of the coarse elements.In particular, the discontinuity boundary is given by y = x

√2 + 0.5, i.e., the discontinuity line intersects

the coarse grid blocks. Similar to the aligned case, various jump magnitudes are considered. These resultsdemonstrate the robustness of our approach for anisotropic fields where h and ε are nearly the same, andthe fluxes that are discontinuous spatial functions.

As for CPU comparisons, we have observed more than 92 percent CPU savings when using MsFEM with-out oversampling. With the oversampling approach, the CPU savings depend on the size of the oversampleddomain. For example, if the oversampled domain size is two times larger than the target coarse block (halfcoarse block extension on each side) we have observed 70 percent CPU savings for 64 × 64 and 80 percentCPU savings for 128 × 128 coarse grid. In general, the computational cost will decrease if the oversampleddomain size is close to the target coarse block size, and this cost will be close to the cost of MsFEM with-out oversampling. Conversely, the error decreases if the size of the oversampled domains increases. In thenumerical examples studied in our paper, we have observed the same errors for the oversampling methodsusing either one coarse block extension or half coarse block extensions. The latter indicates that the leadingresonance error is eliminated by using a smaller oversampled domain. Oversampled domains with one coarseblock extension are previously used in simulations of flow through heterogeneous porous media. As it isindicated in [62], one can use large oversampled domains for simultaneous computations of the several localsolutions. Moreover, parallel computations will improve the speed of the method because MsFEM is wellsuited for parallel computation [62]. For the problems where aε(x, η, ξ) = aε(x)b(η)ξ (see section 5.2 and thenext section for applications) our multiscale computations are very fast because the base functions are builtin the beginning of the computations. In this case, we have observed more than 95 percent CPU savings.

Table 5.2. Relative MsFEM Errors without Oversampling

NL2-norm H1-norm L∞-norm

Error Rate Error Rate Error Rate

32 0.029 0.115 0.0364 0.053 -0.85 0.156 -0.44 0.0534 -0.94

128 0.10 -0.94 0.234 -0.59 0.10 -0.94

Applications of MsFEM to Richards’ equation are presented in [41].


Table 5.3. Relative MsFVEM Errors without Oversampling



32 0.03 0.13 0.0464 0.05 -0.65 0.19 -0.60 0.05 -0.24

128 0.058 -0.19 0.25 -0.35 0.057 -0.19

Table 5.4. Relative MsFEM Errors with Oversampling



32 0.0016 0.036 0.002964 0.0012 0.38 0.019 0.93 0.0016 0.92

128 0.0024 -0.96 0.0087 1.14 0.0026 -0.71

Table 5.5. Relative MsFVEM Errors with Oversampling



32 0.002 0.038 0.00564 0.003 -0.43 0.021 0.87 0.003 0.72

128 0.001 1.10 0.009 1.09 0.001 1.08

Table 5.6. Relative MsFEM Errors for random heterogeneities, spherical variogram, lx = 0.20, lz = 0.02, σ = 1.0

NL2-norm H1-norm L∞-norm hor. flux

Error Rate Error Rate Error Rate Error Rate

32 0.0006 0.0505 0.0025 0.02564 0.0002 1.58 0.029 0.8 0.001 1.32 0.017 0.57

128 0.0001 1 0.016 0.85 0.0005 1 0.011 0.62

Table 5.7. Relative MsFVEM Errors for random heterogeneities, spherical variogram, lx = 0.20, lz = 0.02, σ = 1.0



32 0.0006 0.0515 0.0025 0.02764 0.0002 1.58 0.029 0.81 0.0013 0.94 0.018 0.58

128 0.0001 1 0.016 0.85 0.0005 1.38 0.012 0.58

Table 5.8. Relative MsFEM Errors for random heterogeneities, spherical variogram, lx = 0.20, lz = 0.02, σ = 1.0,aligned discontinuity, jump = exp(1)



32 0.0006 0.0641 0.0020 0.03964 0.0002 1.58 0.0382 0.75 0.0010 1.00 0.027 0.53

128 0.0001 1.00 0.0210 0.86 0.0005 1.00 0.018 0.59

60 Thomas Y. Hou




32 0.0008 0.0817 0.0040 0.06164 0.0004 1.00 0.0493 0.73 0.0023 0.80 0.041 0.57

128 0.0002 1.00 0.0256 0.95 0.0011 1.06 0.025 0.71




32 0.0011 0.1010 0.0068 0.19564 0.0006 0.87 0.0638 0.66 0.0045 0.59 0.109 0.84

128 0.0003 1.00 0.0349 0.87 0.0024 0.91 0.063 0.79

Table 5.11. Relative MsFEM Errors for random heterogeneities, spherical variogram, lx = 0.20, lz = 0.02, σ = 1.0,nonaligned discontinuity, jump = exp(1)



32 0.0006 0.0623 0.0023 0.03564 0.0002 1.58 0.0366 0.77 0.0014 0.72 0.024 0.54

128 0.0001 1.00 0.0203 0.85 0.0006 1.22 0.016 0.59




32 0.0010 0.0785 0.0088 0.05264 0.0003 1.74 0.0440 0.84 0.0052 0.76 0.031 0.75

128 0.0001 1.59 0.0239 0.88 0.0022 1.24 0.017 0.87




32 0.0067 0.1775 0.1000 0.16464 0.0016 2.07 0.0758 1.23 0.0288 1.80 0.077 1.09

128 0.0009 0.83 0.0687 0.14 0.0423 -0.55 0.039 0.98


5.6 Generalizations of MsFEM and some remarks

Next, we present the framework of MsFEM for general equations. Consider

Lεuε = f, (5.35)

where ε is a small scale and Lε : X → Y is an operator. Moreover, we assume that Lε G-converges to L∗ (upto a sub-sequence), where u is a solution of

L∗u = f, (5.36)

(we refer to [89], page 14 for the definition of G-convergence for operators). The objective of MsFEM is toapproximate u in Sh. Denote Sh a family of finite dimensional space such that it possesses an approximationproperty (see [106], [90]) as before. Here h is a scale of computation and h ε. For (5.35) multiscale mapping,EMsFEM : Sh → V hε , will be defined as follows. For each element vh ∈ Sh, vε,h = EMsFEM vh is defined as

Lmapε vε,h = 0 in K, (5.37)

where Lmapε can be, in general, different from Lε and allows us to capture the effects of the small scales.Moreover, the domains different from the target coarse block K can be used in the computations of thelocal solutions. To solve (5.37) one needs to impose boundary and initial conditions. This issue needs to beresolved on a case by case basis, and the main idea is to interpolate vh onto the underlying fine grid. Further,we seek a solution of (5.35) in V hε as follows. Find uh ∈ Sh (consequently uε,h ∈ V hε ) such that

〈Lglobalε uε,h, vh〉 = 〈f, vh〉, ∀vh ∈ Sh, (5.38)

where 〈u, v〉 denotes the duality between X and Y , and Lglobalε can be, in general, different from Lε.For example, for nonlinear elliptic equations we have Lεu = −divaε(x, u,∇u) + a0,ε(x, u,∇u), Lmapε u =divaε(x, η

u,∇u) in K, and Lglobalε = divaε(x, ηu,∇u) + a0,ε(x, η

u,∇u) in K. The convergence of MsFEM isto show that uh → u and uε,h → uε, where uε,h = EMsFEMuh in appropriate space. The correct choices ofLmapε and Lglobalε are the essential part of MsFEM and guarantees the convergence of the method.

In conclusion, we have presented a natural extension of MsFEM to nonlinear problems. This is accom-plished by considering a multiscale map instead of the base functions that are considered in linear MsFEM[62]. Our approaches share some common elements with recently introduced HMM [37], where macroscopicand microscopic solvers are also needed. In general, the finding of “correct” macroscopic and microscopicsolvers is the main difficulty of the multiscale methods. Our approaches follow MsFEM and, consequently,finite element methods constitute its main ingredient. The resonance errors, that arise in linear problemsalso arise in nonlinear problems. Note that the resonance errors are the common feature of multiscale meth-ods unless periodic problems are considered and the solutions of the local problems in an exact period areused. To reduce the resonance errors we use oversampling technique and show that the error can be greatlyreduced by sampling from the larger domains. The multiscale map for MsFEM uses the solutions of thelocal problems in the target coarse block. This way one can sample the heterogeneities of the coarse block.If there is a scale separation and, in addition, some kind of periodicity, one can use the solutions of thesmaller size problems to approximate the multiscale map. Note that a potential disadvantage of periodicityassumption is that the periodicity can act to disrupt large-scale connectivity features of the flow. For theexamples similar to the non-periodic ones considered in this paper, with the use of the smaller size problemsfor approximating the solutions of the local problems, we have found very large errors (of order 50 percent).

6 Multiscale simulations of two-phase immiscible flow in adaptive coordinate

system

Previously, we discussed some applications of MsFEM to two-phase flows. In this section, we explore the useof adaptive coordinate system in multiscale simulations of two-phase porous media flows. In particular, wewould like to present upscaling of transport equations and its coupling to MsFEM.

62 Thomas Y. Hou

As we discussed earlier, the use of global information can improve the multiscale finite element method. Inparticular, the solution of the pressure equation at initial time is used to construct the boundary conditionsfor the basis functions. It is interesting to note that the multiscale finite element methods that employ alimited global information reduces to standard multiscale finite element method in flow-based coordinatesystem. This can be verified directly and the reason behind it is that we have already employed a limitedglobal information in flow-based coordinate system. To achieve high degree of speed-up in two-phase flowcomputations, we also consider the upscaling of transport equation and its coupling to pressure equation.

We would like to derive an upscaled model for the transport equation. We will assume the velocity isindependent of time, λ(S) = 1, and restrict ourselves to the two-dimensional case. Then using the pressure-streamline framework, one obtains

Sεt + vε0f(Sε)p = 0 (6.1)

S(p, ψ, t = 0) = S0,

where ε denotes the small scale, vε0 denotes the Jacobian of the transformation and is positive, and p denotesthe initial pressure. For simplicity, we assume k(x) = k(x)I and we have ∇ψ · ∇p = 0. For deriving upscaledequations, we will first homogenize (6.1) along the streamlines, and then to homogenize across the streamlines.The homogenization along the streamlines can be done following Bourgeat and Mikelic [17] or following Houand Xin [68] and E [36]. The latter uses two-scale convergence theory and we refer to [95] for the resultson homogenization of (6.1) using two-scale convergence theory. We note that the homogenization results ofBourgeat and Mikelic is for general heterogeneities without an assumption on periodicity, and thus, is moreappropriate for problems considered in the paper. Following [17], the homogenization of (6.1) can be easilyderived (see Proposition 3.4 in [17]). Here, we briefly sketch the proof.

For ease of notations, we ignore the ψ dependence of vε0 and Sε, and treat ψ as a parameter. We consider

vε0(p) = v0(p,p

ε).

Moreover, we assume that the domain is a unit interval. Then, for each ψ, it can be shown that Sε(p, ψ, t) →S(p, ψ, t) in L1((0, 1) × (0, T )), where S satisfies

St + v0f(S)p = 0, (6.2)

and where v0 is harmonic average of vε0, i.e.,

1

vε0→ 1

v0weak ∗ in L∞(0, 1).

The proof of this fact follows from Proposition 3.4. of [17].

Following [17] and assuming for simplicity∫ 1

0dηvε0(η) =

∫ 1

0dηv0(η) = 1, we introduce

dXε(p)

dp= vε0(X

ε(p)),dX0(p)

dp= v0(X

0(p)).

Then (Lemma 3.1 of [17]):Xε → X0 in C[0, 1] as ε→ 0. (6.3)

Consequently,∫ T

0

∫ 1

0

|Sε(p, τ) − S(p, τ)|dpdτ =

∫ T

0

∫ 1

0

|Sε(Xε(p), τ) − S(Xε(p), τ)|vε0(Xε(p))dpdτ ≤∫ T

0

∫ 1

0

|Sε(Xε(p), τ) − S(X0(p), τ)|vε0(Xε(p))dpdτ+

∫ T

0

∫ 1

0

|S(Xε(p), τ) − S(X0(p), τ)|vε0(Xε(p))dpdτ ≤∫ T

0

∫ 1

0

|Sε(Xε(p), τ) − S(X0(p), τ)|dpdτ +

∫ T

0

∫ 1

0

|S(Xε(p), τ) − S(X0(p), τ)|dpdτ

(6.4)


The first term on the right hand side of (6.4) converges to zero because Sε(Xε(p), τ) and S(X0(p), τ) satisfythe same equation ut + f(u)p = 0, however, with the following initial conditions Sε(Xε(p), t = 0) = S0 Xε,

S(X0(p), τ) = S0 X0. Because of (6.3) and comparison principle

∫ T

0

∫ 1

0

|Sε(Xε(p), τ) − S(X0(p), τ)|dpdτ ≤ C

∫ 1

0

|S0 Xε − S0 X0|dp,

the first term converges to zero. The convergence of the second term for each ψ follows from the argumentin [17] (page 368) using Lebesgue’s dominated convergence theorem.

Next, we provide a convergence rate (see also [95]) of the fine saturation Sε to the homogenized limit Sas ε→ 0.

Theorem 61 Assume that vε0(p) is bounded uniformly

C−1 ≤ vε0(p,p

ε) ≤ D.

Denote by F (t, T ) the solution to St + f(S)T = 0. The solution S of (6.2) converges to Sε (assuming initialconditions that don’t depend on the fast scale) at a rate given by

‖Sε − S‖∞ ≤ Gε,

when F remains Lipschitz for all time, and

‖Sε − S‖n ≤ Gε1/n,

when F develops at most a finite number of discontinuities.

Proof. First, we note that the velocity bound implies that C−1 ≤ v0(p) ≤ D, uniformly in ψ, ζ. We transformthe equations for Sε (6.1) and S (6.2) to the time of flight variable defined by

dT ε

dp = 1vε0(p,ψ)

T ε(0) = 0for Sε and

dTdp = 1

v(p,ψ,ψε)

T (0) = 0for S.

Both equations reduce toSt + f(S)T = 0.

The solution to this equation is F (t, T ). Since the initial condition does not depend on ε neither doesF . Then S = F (t, T ε(P,Ψ)), S = F (t, T (P,Ψ)). Using these expressions for the saturation we can obtainthe desired estimates by following the same steps as in the linear case. When F remains Lipschitz forall times we can easily obtain a pointwise estimate in terms of the Lipschitz constant M ‖Sε − S‖∞ =‖F (t, T ε) − F (t, T )‖∞ ≤ M‖T ε − T‖∞ ≤ Gε. Otherwise we will need the time of flight bound that wederived for the linear flux that reduces here to

|T ε(P ) − T (P )| ≤ 2Cε. (6.5)

We will divide the domain in regions where F is Lipschitz with constantM in the second variable, denotedby A2, and shock regions, denoted by A1, and estimate the difference of Sε and S in each region separately.To fix the notation, let that there be n discontinuities in F (t, ·) of magnitude less than ∆F , which does nothave to be small, at T = Tii=1,...,n. We will denote the thin strips of width 2Cε around the discontinuitieswith A1

A1 = T such that |T − Ti| ≤ 2Cε, for some i = 1, . . . , nand with A2 its complement. We selected the width of the strip based on (6.5), so that for any point P , ifT ε(P ) /∈ A1, then T ε(P ) and T (P ) are on the same side of any jump Ti. When T ε(P ) ∈ A2, F is Lipschitzin the region between T ε and T , and we can show

∫

A2(Sε − S)2dpdψ =

∫

A2(F (t, T ε) − F (t, T ))2dpdψ ≤ M2‖T ε − T‖2

∞|T ε(A2)−1|

≤ N2ε2|T ε(A2)−1|,

64 Thomas Y. Hou

where we used the time of flight bound (6.5). By |T ε(A2)−1| we denoted the image of A2 under the inverse

of T ε(P ). Inside the strip A1, even though Sε and S differ by an O(1) quantity we can use the smallness ofthe area of the strip to make the L2 norm of their difference small

∫

A1(Sε − S)2dpdψ =

∫

A2(F (t, T ε) − F (t, T ))2dpdψ ≤ (∆S +Nε)2|T ε(A1)

−1|≤ (∆S +Nε)24CDnε.

We estimated the area |T ε(A1)−1| by using the definition of A1 and the fact that the Jacobian of the

transformation T ε(P )−1 is vε0 and is bounded uniformly in p, ψ. Putting together the two estimates forregions A1 and A2 we obtain ‖Sε− S‖2 ≤ Gε1/2. Estimates in terms of the other Lp norms follow similarly.

The homogenized operator given by (6.2) still contains variation of order ε through the fast variable ψε ,

however there it does not contain any derivatives in that variable. Its dependence on ψε is only parametric.

We can homogenize the dependence of the partially homogenized operator on ψε and arrive at a homogenized

operator that is independent of the small scale. In the latter case, we will only obtain weak convergenceof the partially homogenized solution. When we homogenized along the streamlines, the resulting equationwas of hyperbolic type like the original equation. In a seminal and celebrated paper, Tartar [99] showedthat homogenization across streamlines leads to transport with the average velocity plus a time-dependentdiffusion term, referred to as macrodispersion, a physical phenomenon that was not present in the originalfine equation. In particular, if the velocity field does not depend on p inside the cells, that is, v(ψ, ψε ), then

the homogenized solution, S, (weak∗ limit of S, which will be denoted by S), satisfies

St + v0Sp =

∫ t

0

∫

Spp(p− λ(t− τ), ψ, τ)dµψε(λ)dτ. (6.6)

Here, dνψε

the Young measure associated with the sequence v0(ψ, ·) and dµψε

is a Young measure that

satisfies(

∫ dνψε(λ)

s2πiq + λ

)−1

=s

2πiq+ v0 −

∫ dµψε(λ)

s2πiq + λ

.

We have denoted by v0 the weak limit of the velocity. This equation has no dependence on the small scale andwe consider it to be the full homogenization of the fine saturation equation. Efendiev and Popov [47] haveextended this method for the Riemann problem in the case of nonlinear flux. Note that the homogenizationacross streamlines provides a weak limit of partially homogenized solution. Because the original solution S ε

strongly converges to partially homogenized solution for each ψ, it can be easily shown that Sε → S weakly.We omit this proof here.

In numerical simulations, it is difficult to use (6.6) as a homogenized operator, and often a second orderapproximation of this equation is used. These approximate equations can be also derived using perturbationanalysis. In particular, using the higher moments of the saturation and the velocity, one can model themacrodispersion. In the context of two-phase flow this idea was introduced by Efendiev, Durlofsky, and Lee[46], [45], Chen and Hou [25] and Hou et al., [61]. In our case, the computation of the macrodispersion ismuch simpler because the transport equations have been already averaged along the streamlines, and thuswe will be applying perturbation technique to one dimensional problem.

We expand S, v0 (following [46]) as an average over the cells in the pressure-streamline frame and thecorresponding fluctuations

S = S(p, ψ, t) + S′(p, ψ, ζ, t)v0 = v0(p, ψ, t) + v′0(p, ψ, ζ, t).

(6.7)

We will derive the homogenized equation for f(S) = S. Averaging equations (6.2) with respect to ψ we findan equation for the mean of the saturation

St + v0Sp + v′0S′p = 0.


An equation for the fluctuations is obtained by subtracting the above equation from (6.2)

S′t + (v0 − v0)Sp + v0S

′p − v′0S

′p = 0.

Together, the equations for the saturation are

St + v0Sp + v′0S′p = 0 (6.8)

S′t + v′0Sp + v0S

′p − v′0S

′p = 0.

We can consider the second equation to be the auxiliary (cell) problem and the first equation to be theupscaled equation. We note that the cell problem for a hyperbolic equation is O(1) whereas for an elliptic itis O(ε). We can obtain an approximate numerical method by solving the cell problem only near the shockregion in space time, where the macrodispersion term is largest. In that case it is best to diagonalize theseequations by adding the first to the second one

St + v0Sp = −v′0(Sp − Sp)

St + v0Sp = 0.

Compared to (6.8), it has fewer forcing terms and no cross fluxes, which leads to a numerical method withless numerical diffusion that is easier to implement .

6.1 Numerical Averaging across Streamlines

The derivation in the previous sections contained no approximation. In this section, we follow the sameidea as in the derivation to solve the equation for the fluctuations along the characteristics, but with thepurpose of deriving an equation on the coarse grid. To achieve this, we will not perform analytical upscalingin the sense of deriving a continuous upscaled equation as in the previous section. We will first discretizethe equation with a finite volume method in space and then upscaled the resulting equation. Our upscaledequation will therefore be dependent on the numerical scheme.

We use the same definition for the average saturation and the fluctuations as in (6.7) and follow the samesteps until equation (6.8). We discretize the macrodispersion term in the equation for the average saturation

v′0S′p =

v′0S′i+1 − v′0S

′i

∆p+O(∆p).

A superscript ·i refers to a discrete quantity defined at the center of the conservation cell. Instead of solvingthe equation for the fluctuations on the fine characteristics as before, which would lead to a fine grid algorithm,we solve it on the coarse characteristics defined by

dP

dt= v0, with P (p, 0) = p.

Compared to the equation that we obtained in the previous section for S ′, this equation for S′ has an extraterm, which appears second

S′ = −∫ t

0

(

v′0(P (p, τ), ψ)Sp(P (p, τ), ψ, τ) + v′0(P (p, τ), ψ)S′p(P (p, τ), ψ, τ) + v′0S

′p))

dτ.

The second term is second-order in fluctuating quantities, and we expect it to be smaller than the first termso we neglect it. As before, we multiply by v′0 and average over ψ to find

v′0S′ = −

∫ t

0

v′0v0(P (p, τ), ψ)Sp(P (p, τ), ψ, τ)dτ.

66 Thomas Y. Hou

In this form at time t it is necessary to know information about the past saturation in (0, t) to computethe future saturation. Following [46], it can be easily shown that Sp(P (p, τ) depends weakly on time, inthe sense that the difference between Sp(P (p, τ) and Sp(P (p, t) is of third-order in fluctuating quantities.Therefore we can take Sp(P (p, τ) out of the time integral to find

v′0S′ = −

∫ t

0

v′0v′0(P (p, τ), ψ)dτSp.

The term inside the time integral is the covariance of the velocity field along each streamline. The macrodis-persion in this form can be computed independent of the past saturation.

The nonlinearity of the flux function introduces an extra source of error in the approximation. We expandf(S) near S (cf. [45]) and keep only the first term

S = S(p, ψ, t) + S′(p, ψ, ζ, t)v0 = v0(p, ψ, t) + v′0(p, ψ, η, t)f(S) = f(S) + fS(S)S′ +O(S′2)f(S)p = fS(S)Sp + f(S)S′ + . . .

(6.9)

This approximation is not accurate near the shock because S ′ is not small near sharp fronts. The regionnear the shock is important because the macrodispersion is large. Due to the dependence of the jump inthe saturation on the mobility we expect this approximation to be better for lower mobilities. Neverthelessthis approximation works well in practice. For more accuracy, it is also possible to retain more terms in theTaylor expansion. We will show that in realistic examples these higher-order terms are not important in oursetting.

Using these definitions we derive the following equations for the average saturation and the fluctuations(see [95] for more details)

St + v0f(S)p + v′0(fS(S)S′)p = 0 (6.10)

S′t + v′0fS(S)Sp + v0fS(S)S′

p − v′0S′p = 0.

The macrodispersion is discretized as

v′0(fS(S)S′)p =v′0fS(S)S′

i+1− v′0fS(S)S′

i

∆p+O(∆p).

We solve the second equation on the coarse characteristics defined by

dP

dt= v0fS(S), with P (p, 0) = p

and form the terms that appear in the macrodispersion

v′0fS(S)S′ = −∫ t

0

v′0fS(S)v′0(P (p, τ), ψ)fS(S(P (p, τ), ψ, τ))Sp(P (p, τ), ψ, τ)dτ.

As before we have dropped terms that are second-order in fluctuating quantities. It can be shown (see [95])that fS(S(P (p, τ), ψ, τ))Sp(P (p, τ), ψ, τ) does not vary significantly along the streamlines and it can be takenout of the integration in time:

v′0fS(S)S′ = −∫ t

0

v′0v′0(P (p, τ), ψ)dτfS(S)2Sp. (6.11)

This expression is similar to the one obtained in the linear case, however the macrodispersion depends onthe past saturation through the equation for the coarse characteristics.


Even though the macrodispersion depends on the past saturation it is possible to compute it incrementallyas it is done in [45]. Given its value D(t) at time t we compute the values at t+∆t using the macrodispersionat the previous time

D(t+ ∆t) =

∫ t+∆t

0

. . . dτ =

∫ t

0

. . . dτ +

∫ t+∆t

t

. . . dτ.

This is possible because in the derivation for the approximate expression for the macrodispersion we tookthe terms that depend on S(τ) outside the time integration. The integrand, the average covariance of thevelocity field along the streamlines, needs to be computed only once at the beginning. Then updating themacrodispersion takes O(n2) computations, as many as it takes to update S.

6.2 Numerical Results

In this section, we first show representative simulation results for λ(S) = 1 for flux functions f(S) = S andnonlinear f(S) with viscosity ratio µo/µw = 5. For such setting, the pressure and saturation equations aredecoupled and we can investigate the accuracy of saturation upscaling independently from the pressure up-scaling. At the end of the section we will present numerical results for two-phase flow. We consider two typeof permeability fields. The first type includes a permeability field generated using two-point geostatistics withcorrelation lengths lx = 0.3, lz = 0.03 and σ2 = 1.5 (see Figure 6.1, left). The second type of permeabilityfields correspond to a channelized system, and we consider two examples. The first example (middle figure ofFigure 6.1) is a synthetic channelized reservoir generated using both multi-point geostatistics (for the chan-nels) and two-point geostatistics (for permeability distribution within each facies). The second channelizedsystem is one of the layers of the benchmark test (representing the North Sea reservoir), the SPE compar-ative project [27] (upper Ness layers). These permeability fields are highly heterogeneous, channelized, anddifficult to upscale. Because the permeability fields are highly heterogeneous, they are refined to 400 × 400in order to obtain accurate comparisons.

Simulation results will be presented for saturation snapshots as well as the oil cut as a function ofpore volume injected (PVI). Note that the oil cut is also referred to as the fractional flow of oil. The oilcut (or fractional flow) is defined as the fraction of oil in the produced fluid and is given by qo/qt, whereqt = qo + qw, with qo and qw being the flow rates of oil and water at the production edge of the model.In particular, qw =

∫

∂Ωoutf(S)v · ndl, qt =

∫

∂Ωoutv · ndl, and qo = qt − qw, where ∂Ωout is the outer flow

boundary. We will use the notation Ω for total flow qt and F for fractional flow qo/qt in numerical results.

Pore volume injected, defined as PV I = 1Vp

∫ t

0qt(τ)dτ , with Vp being the total pore volume of the system,

provide a dimensionless time for the displacement.

When using multiscale finite element methods for two-phase flow, one can update the basis functions nearthe sharp fronts. Indeed, sharp fronts modify the local heterogeneities and this can be taken into accountby re-solving the local equations, (4.34), for basis functions. If the saturation is smooth in the coarse block,it can be approximated by its average in (4.34), and consequently, the basis functions do not needed to beupdated. It can be shown that this approximation yields first-order errors (in terms of coarse mesh size). Inour simulations, we have found only a slight improvement when the basis functions are updated, thus thenumerical results for the MsFVEM presented in this paper do not include the basis function update nearthe sharp fronts. Since a pressure-streamline coordinate system is used the boundary conditions are givenby P = 1, S = 1 along the p = 1 edge and P = 0 along the p = 0 edge, and no flow boundary condition onthe rest of the boundaries.

For the upscaled saturation equation, which is a convection-diffusion equation, we need to observe an

extra CFL-like condition to obtain a stable numerical scheme ∆t ≤ ∆p2

2ν , where ν is the diffusivity. In our case

the diffusivity is∫

cell

∫ t

0 v′0(p(τ), ψ)v′0(p, ψ)dτdψ. If the macrodispersion is large this can be a very restrictive

condition. To remedy this, we used an implicit discretization for the macrodispersion. This is straightforwardsince the problem is one-dimensional. The resulting system was solved by a tridiagonal solver very fast. Sincethe order of the highest derivative in the equation has increased, we require extra boundary conditions. Forthe computation of the macrodispersion term, we impose no flux on both boundaries of the domain.

68 Thomas Y. Hou

−3

−2

−1

0

1

2

3

−1

0

1

2

3

4

5

6

7

8

9

−4

−2

0

2

4

6

8

Fig. 6.1. Permeability fields used in the simulations. Left - permeability field with exponential variogram, middle -synthetic channelized permeability field, right - layer 36 of SPE comparative project [27]

In the upscaled algorithm, a moving mesh is used to concentrate the points of computation near thesharp front. Since the saturation equation is one dimensional in the pressure-streamline coordinates, theimplementation of the moving mesh is straightforward and efficient. For the details we refer to [95]. Wecompare the saturation right before the breakthrough time so that the shock front is largest. For thiscomparison we also average the fine saturation over the coarse blocks, since the upscaled model is definedon a coarser grid. In Figures 6.2 6.3, we plot the saturation for linear and nonlinear (with µo/µw = 5) f(S).As we see in both cases, we have very accurate representation of the saturation profile.

We proceed with a quantitative description of the error. We will distinguish between two sources of errors.We will refer to the difference between the upscaled and the exact equation as the upscaling or modelingerror and to the difference between the solution of continuous upscaled equations and the solution to thenumerical scheme as the discretization error. We will refer to the difference between the solutions of thecontinuous fine equations and the numerical scheme of the upscaled equations as the total error. To separatethe upscaling error from the total error we will solve the upscaled equations on the fine grid, which is thegrid on which we solve to the fine equation. We will also solve them on the coarse grid to compute the totalerror. The errors are computed in the p, ψ frame and are relative errors. We display the upscaling erroragainst the number of coarse cells for the computations of the previous section in Tables 6.14, 6.15, 6.16. Aswe see from this table that upscaling using macrodispersion decreases the upscaling errors. We also see thatthe effects of macrodispersion are more significant in the case of linear flux when the jump discontinuity inthe saturation profile is larger.

Table 6.14. Upscaling error for permeability generated using two-point geostatistics

LINEAR FLUX 25x25 50x50 100x100 200x200

L1 error of S 0.0021 6.57 × 10−4 2.15 × 10−4 8.75 × 10−5

L1 error of S with macrodispersion 0.115 0.0696 0.0364 0.0135

L1 error of S fine without macrodispersion 0.1843 0.0997 0.0505 0.0191

NONLINEAR FLUX 25x25 50x50 100x100 200x200

L1 error of S 0.0023 8.05 × 10−4 2.89 × 10−4 1.29 × 10−4



In Tables 6.17, 6.18, 6.19, we show the total error, that is, the modeling and discretization error whenwe use a moving mesh to solve the saturation equation. It is interesting that the convergence of S to S isobserved even though the upscaling error is larger than the numerical error of the fine solution, which is0.02 for the linear flux and 0.002 for the nonlinear flux in the L1 norm, as mentioned before. The reason isthat the location of the moving mesh points was selected so that the points are as dense near the shock asthe fine solution using the parameter hmin. This was done to observe the upscaling error clearly and also


0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

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0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fig. 6.2. Saturation snapshots for variogram based permeability field (top) and synthetic channelized permeabilityfield (bottom). Linear flux is used. Left figures represent the upscaled saturation plots and the right figures representthe fine-scale saturation plots.

Table 6.15. Upscaling error for for synthetic channelized permeability field

LINEAR FLUX 25x25 50x50 100x100 200x200

L1 error of S 0.0222 0.0171 0.0122 0.0053




L1 error of S 0.0147 0.0105 0.0075 0.0040



70 Thomas Y. Hou

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

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0.2

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0.4

0.5

0.6

0.7

0.8

0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fig. 6.3. Saturation snapshots for variogram based permeability field (top) and synthetic channelized permeabilityfield (bottom). Nonlinear flux is used. Left figures represent the upscaled saturation plots and the right figuresrepresent the fine-scale saturation plots.

Table 6.16. Upscaling error for SPE 10, layer 36

LINEAR FLUX 25x25 50x50 100x100 200x200

L1 error of S 0.0128 0.0093 0.0072 0.0042




L1 error of S 0.0089 0.0064 0.0054 0.0033




to have similar CFL constraints on the time step, which allows a clean comparison of computational times.We compare the require CPU times in Table 6.20. We note that it took 26 units of time to interpolate onequantity from the Cartesian to the pressure-streamline frame. The upscaled solutions were computed on a25 × 25 grid and the fine solution was computed on a 400 × 400 grid so we expect the S computations tobe 256 times or more faster. The extra gain comes from a less restrictive CFL condition since we use anaveraged velocity. The computations in the Cartesian frame are much slower.

Table 6.17. Total error for permeability field generated using two-point geostatistics

LINEAR FLUX 25x25 50x50 100x100 200x200

L1 upscaling error of S 0.0021 6.57 × 10−4 2.15 × 10−4 8.75 × 10−5

L1 error of S computed on coarse grid 0.0185 0.0062 0.0019 0.0015

L1 upscaling error of S 0.115 0.0696 0.0364 0.0135

L1 error of computed on coarse grid 0.139 0.0779 0.0390 0.0144


L1 upscaling error of S 0.0023 8.05 × 10−4 2.89 × 10−4 1.29 × 10−4

L1 error of S computed on coarse grid 0.0268 0.0099 0.0027 9.38 × 10−4



Table 6.18. Total error for synthetic channelized permeability field

LINEAR FLUX 25x25 50x50 100x100 200x200










The application of the proposed method to two-phase immiscible flow can be performed using the implicitpressure and explicit saturation (IMPES) framework. This procedure consists of computing the velocity andthen using the velocity field in updating the saturation field. When updating the saturation field, we considerthe velocity field to be time independent and we can use our upscaling procedure at each IMPES time step.First, we note that in the proposed method, the mapping is done between the current pressure-streamlineand initial pressure-streamline. This mapping is nearly the identity for the cases when µo > µw. In Figure6.4, we plot the level sets of the pressure and streamfunction at time t = 0.4 in a Cartesian coordinatesystem (left plot) and in the coordinate system of the initial pressure and streamline (right plot). Clearly,the level sets are much smoother in initial pressure-streamline frame compared to Cartesian frame. This alsoexplains the observed convergence of upscaling methods as we refined the coarse grid. In Figure 6.5, we plotthe saturation snapshots right before the breakthrough. In Figure 6.6, the fractional flow is plotted. Again,the moving mesh algorithm is used to track the front separately. The convergence table is presented in Table6.21. We see from this table that the errors decrease as first order which indicates that the pressure andsaturation is smooth functions of initial pressure and streamline.

72 Thomas Y. Hou

Table 6.19. Total error for SPE10 layer 36

LINEAR FLUX 25x25 50x50 100x100 200x200










Table 6.20. Computational cost

fine x.y fine p, ψ S S

layered, linear flux 5648 257 9 1

layered, nonlinear flux 14543 945 28 4

percolation, linear flux 8812 552 12 1

percolation, nonlinear flux 23466 579 12 1

SPE10 36, linear flux 40586 1835 34 2

SPE10 36, nonlinear flux 118364 7644 25 2

Fig. 6.4. Left: Pressure and streamline function at time t = 0.4 in Cartesian frame. Right: pressure and streamlinefunction at time t = 0.4 in initial pressure-streamline frame.

7 Conclusions

In these lecture notes, we reviewed some of the recent advances in developing systematic multiscale methodswith particular emphasis on multiscale finite element methods and their applications to fluid flows in hetero-geneous porous media. In particular, the local approaches and their convergence properties for various flowproblems are discussed. Moreover, improved subgrid capturing techniques through a judicious choice of localboundary conditions or through oversampling techniques or through the use of limited global informationare reviewed. Other topics, such as homogenization, the sampling techniques in numerical homogenization,and multiscale simulations of two-phase flows in heterogeneous porous media are also presented. Althoughthe results presented in this paper are encouraging, there is scope for further exploration. These include


0.1

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0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fig. 6.5. Left: Saturation plot obtained using coarse-scale model. Right: The fine-scale saturation plot. Both plotsare on coarse grid. Variogram based permeability field is used. µo/µw = 5.

0 0.5 1 1.5 2 2.5 3 3.5 4pvi

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time

fra

ctio

na

l flo

w r

ate

fineupscaled

Fig. 6.6. Comparison of fractional flow for coarse- and fine-scale models. Variogram based permeability field is used.µo/µw = 5.

Table 6.21. Convergence of the upscaling method for two-phase flow for variogram based permeability field

with S 50x50 100x100 200x200

L2 pressure error at t =3Tfinal

40.0014 0.007 0.004

L2 velocity error at t =3Tfinal

40.0235 0.0137 0.0072

L1 saturation error t = Tfinal 0.0105 0.0052 0.0027

with S 50x50 100x100 200x200

L2 pressure error at t =3Tfinal

40.0046 0.0021 0.0008

L2 velocity error at t =3Tfinal

40.0530 0.0335 0.0246

L1 saturation error t = Tfinal 0.0546 0.0294 0.0134

74 Thomas Y. Hou

the development and mathematical analysis of efficient numerical homogenization techniques for nonlin-ear convection-diffusion equations with various Peclet numbers (e.g., convection dominated), inexpensiveapproximations of multiscale basis functions, further exploration of accurate boundary conditions basedon local multiscale solutions, the use of limited global information for nonlinear problems, development ofadaptive criteria for multiscale basis functions (selection of coarse grid), applications of MsFEM to moremulti-phase/multi-component porous media flows, and etc.

References

1. J. Aarnes, On the use of a mixed multiscale finite element method for greater flexibility and increased speed orimproved accuracy in reservoir simulation, SIAM MMS, 2 (2004), pp. 421–439.

2. J. Aarnes, Y. R. Efendiev, and L. Jiang, Analysis of multiscale finite element methods using global informationfor two-phase flow simulations. submitted.

3. J. Aarnes and T. Y. Hou An Efficient Domain Decomposition Preconditioner for Multiscale Elliptic Problemswith High Aspect Ratios, Acta Mathematicae Applicatae Sinica, 18 (2002), 63-76.

4. R. A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], NewYork-London, 1975. Pure and Applied Mathematics, Vol. 65.

5. T. Arbogast, Numerical Subgrid Upscaling of Two-Phase Flow in Porous Media, in Numerical treatment ofmultiphase flows in porous media, Z. Chen et al., eds., Lecture Notes in Physics 552, Springer, Berlin, 2000, pp.35-49.

6. T. Arbogast, Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase Darcyflow, Comput. Geosci., 6 (2002), pp. 453–481. Locally conservative numerical methods for flow in porous media.

7. T. Arbogast and K. Boyd, Subgrid upscaling and mixed multiscale finite elements. to appear in SIAM Num. Anal.8. M. Avellaneda, T. Y. Hou and G. Papanicolaou, Finite Difference Approximations for Partial Differential Equa-

tions with Rapidly Oscillating Coefficients, Mathematical Modelling and Numerical Analysis, 25 (1991), 693-710.9. M. Avellaneda and F.-H. Lin, Compactness methods in the theory of homogenization, Comm. Pure Appl.

Math., 40 (1987), pp. 803–847.10. I. Babuska, U. Banerjee, and J. E. Osborn, Survey of meshless and generalized finite element methods: A unified

approach, Acta Numerica, 2003, pp. 1-125.11. I. Babuska, G. Caloz, and E. Osborn, Special Finite Element Methods for a Class of Second Order Elliptic

Problems with Rough Coefficients, SIAM J. Numer. Anal., 31 (1994), 945-981.12. I. Babuska and J. M. Melenk, The partition of unity method, Internat. J. Numer. Methods Engrg., 40 (1997),

pp. 727–758.13. I. Babuska and E. Osborn, Generalized Finite Element Methods: Their Performance and Their Relation to Mixed

Methods, SIAM J. Numer. Anal., 20 (1983), 510-536.14. I. Babuska and W. G. Szymczak, An Error Analysis for the Finite Element Method Applied to Convection-

Diffusion Problems, Comput. Methods Appl. Math. Engrg, 31 (1982), 19-42.15. A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Volume 5 of

Studies in Mathematics and Its Applications, North-Holland Publ., 1978.16. A. Bourgeat, Homogenized Behavior of Two-Phase Flows in Naturally Fractured Reservoirs with Uniform Frac-

tures Distribution, Comp. Meth. Appl. Mech. Engrg, 47 (1984), 205-216.17. A. Bourgeat and A. Mikelic, Homogenization of two-phase immiscible flows in a one-dimensional porous medium,

Asymptotic Anal., 9 (1994), pp. 359–380.18. M. Brewster and G. Beylkin, A Multiresolution Strategy for Numerical Homogenization, ACHA, 2(1995), 327-349.19. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer–Verlag, Berlin – Heidelberg – New-

York, 1991.20. F. Brezzi and A. Russo, Choosing Bubbles for Advection-Diffusion Problems, Math. Models Methods Appl. Sci,

4 (1994), 571-587.21. F. Brezzi, L. P. Franca, T. J. R. Hughes and A. Russo, b =

R

g, Comput. Methods in Appl. Mech. and Engrg.,145 (1997), 329-339.

22. J. E. Broadwell, Shock Structure in a Simple Discrete Velocity Gas, Phys. Fluids, 7 (1964), 1243-1247.23. T. Carleman, Problems Mathematiques dans la Theorie Cinetique de Gaz, Publ. Sc. Inst. Mittag-Leffler, Uppsala,

1957.24. H. Ceniceros and T. Y. Hou, An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions. J.

Comput. Phys., 172 (2001), 609-639.


25. Z. Chen and T. Y. Hou, A mixed multiscale finite element method for elliptic problems with oscillating coefficients,Math. Comp., 72 (2002), pp. 541–576 (electronic).

26. A. J. Chorin, Vortex Models and Boundary Layer Instabilities, SIAM J. Sci. Statist. Comput., 1 (1980), 1-21.

27. M. Christie and M. Blunt, Tenth SPE comparative solution project: A comparison of upscaling techniques, SPEReser. Eval. Eng., 4 (2001), pp. 308–317.

28. M. E. Cruz and A. Petera, A Parallel Monte-Carlo Finite Element Procedure for the Analysis of MulticomponentRandom Media, Int. J. Numer. Methods Engrg, 38 (1995), 1087-1121.

29. J. E. Dendy, J. M. Hyman, and J. D. Moulton, The Black Box Multigrid Numerical Homogenization Algorithm,J. Comput. Phys., 142 (1998), 80-108.

30. C. V. Deutsch and A. G. Journel, GSLIB: Geostatistical software library and user’s guide, 2nd edition,Oxford University Press, New York, 1998.

31. M. Dorobantu and B. Engquist, Wavelet-based Numerical Homogenization, SIAM J.Numer. Anal., 35 (1998),540-559.

32. J. Douglas, Jr. and T.F. Russell, Numerical Methods for Convection-dominated Diffusion Problem Based onCombining the Method of Characteristics with Finite Element or Finite Difference Procedures, SIAM J. Numer.Anal. 19 (1982), 871–885.

33. L. J. Durlofsky, Numerical Calculation of Equivalent Grid Block Permeability Tensors for Heterogeneous PorousMedia, Water Resour. Res., 27 (1991), 699-708.

34. L.J. Durlofsky, R.C. Jones, and W.J. Milliken, A Nonuniform Coarsening Approach for the Scale-up of Displace-ment Processes in Heterogeneous Porous Media, Adv. Water Resources, 20 (1997), 335–347.

35. B. B. Dykaar and P. K. Kitanidis, Determination of the Effective Hydraulic Conductivity for HeterogeneousPorous Media Using a Numerical Spectral Approach: 1. Method, Water Resour. Res., 28 (1992), 1155-1166.

36. W. E, Homogenization of linear and nonlinear transport equations, Comm. Pure Appl. Math., XLV (1992),pp. 301–326.

37. W. E and B. Engquist, The heterogeneous multi-scale methods, Comm. Math. Sci., 1(1) (2003), pp. 87–133.

38. W. E and T. Y. Hou, Homogenization and Convergence of the Vortex Method for 2-D Euler Equations withOscillatory Vorticity Fields, Comm. Pure and Appl. Math., 43 (1990), 821-855.

39. Y. R. Efendiev, Multiscale Finite Element Method (MsFEM) and its Applications, Ph. D. Thesis, Applied Math-ematics, Caltech, 1999.

40. Y. Efendiev, V. Ginting, T. Hou, and R. Ewing, Accurate multiscale finite element methods for two-phase flowsimulations. to appear in Comp. Physics.

41. Y. Efendiev, T. Hou, and V. Ginting, Multiscale finite element methods for nonlinear problems and their appli-cations, Comm. Math. Sci., 2 (2004), pp. 553–589.

42. Y. R. Efendiev, T. Y. Hou, and X. H. Wu, Convergence of A Nonconforming Multiscale Finite Element Method,SIAM J. Numer. Anal., 37 (2000), 888-910.

43. Y. Efendiev and A. Pankov, Homogenization of nonlinear random parabolic operators, Advances in DifferentialEquations, vol. 10,Number 11, 2005, pp., 1235-1260

44. Y. Efendiev and A. Pankov, Numerical homogenization of nonlinear random parabolic operators, SIAM MultiscaleModeling and Simulation, 2(2) (2004), pp. 237–268.

45. Y. R. Efendiev and L. J. Durlofsky, Numerical modeling of subgrid heterogeneity in two phase flow simulations,Water Resour. Res., 38(8) (2002), p. 1128.

46. Y. R. Efendiev, L. J. Durlofsky, S. H. Lee, Modeling of Subgrid Effects in Coarse-scale Simulations of Transportin Heterogeneous Porous Media, WATER RESOUR RES, 36 (2000), 2031-2041.

47. Y. R. Efendiev and B. Popov, On homogenization of nonlinear hyperbolic equations, Communications on Pureand Applied Analysis, 4(2) (2005), pp. 295–309.

48. B. Engquist, Computation of Oscillatory Solutions to Partial Differential Equations, in Proc. Conference onHyperbolic Partial Differential Equations, Carasso, Raviart, and Serre, eds, Lecture Notes in Mathematics No.1270, Springer-Verlag, 10-22, 1987.

49. B. Engquist and T. Y. Hou, Computation of Oscillatory Solutions to Hyperbolic Equations Using Particle Methods,Lecture Notes in Mathematics No. 1360, Anderson and Greengard eds., Springer-Verlag, 68-82, 1988.

50. B. Engquist and T. Y. Hou, Particle Method Approximation of Oscillatory Solutions to Hyperbolic DifferentialEquations, SIAM J. Numer. Anal., 26 (1989), 289-319.

51. B. Engquist and H. O. Kreiss, Difference and Finite Element Methods for Hyperbolic Differential Equations,Comput. Methods Appl. Mech. Engrg., 17/18 (1979), 581-596.

52. B. Engquist and E.D. Luo, Convergence of a Multigrid Method for Elliptic Equations with Highly OscillatoryCoefficients, SIAM J. Numer. Anal., 34 (1997), 2254-2273.

76 Thomas Y. Hou

53. R. Eymard, T. Gallouet, and R. Herbin, Finite volume methods, in Handbook of numerical analysis, Vol. VII,Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, 713–1020.

54. R. Fetecau and T. Y. Hou, A Modified Particle Method for Semi-Linear Hyperbolic Systems with OscillatorySolutions, Methods and Applications of Analysis, 11 (2004), 573-604.

55. J. Fish and K.L. Shek, Multiscale Analysis for Composite Materials and Structures, Composites Science andTechnology: An International Journal, 60 (2000), 2547-2556.

56. J. Fish and Z. Yuan, Multiscale enrichment based on the partition of unity, International Journal for NumericalMethods in Engineering, 62, (2005), 1341–1359.

57. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, Berlin, NewYork, 2001.

58. J. Glimm, H. Kim, D. Sharp, and T. Wallstrom A Stochastic Analysis of the Scale Up Problem for Flow in PorousMedia, Comput. Appl. Math., 17 (1998), 67-79.

59. T. Y. Hou, Homogenization for Semilinear Hyperbolic Systems with Oscillatory Data, Comm. Pure and Appl.Math., 41 (1988), 471-495.

60. T. Hou, X. Wu, and Y. Zhang, Removing the cell resonance error in the multiscale finite element method via apetrov-galerkin formulation, Communications in Mathematical Sciences, 2(2) (2004), 185–205.

61. T. Y. Hou, A. Westhead, and D. P. Yang, A framework for modeling subgrid effects for two-phase flows in porousmedia. to appear in SIAM Multiscale Modeling and Simulation.

62. T. Y. Hou and X. H. Wu, A Multiscale Finite Element Method for Elliptic Problems in Composite Materials andPorous Media, J. Comput. Phys., 134 (1997), 169-189.

63. T. Y. Hou and X. H. Wu, A Multiscale Finite Element Method for PDEs with Oscillatory Coefficients, Proceedingsof 13th GAMM-Seminar Kiel on Numerical Treatment of Multi-Scale Problems, Jan 24-26, 1997, Notes onNumerical Fluid Mechanics, Vol. 70, ed. by W. Hackbusch and G. Wittum, Vieweg-Verlag, 58-69, 1999.

64. T. Y. Hou, X. H. Wu, and Z. Cai, Convergence of a Multiscale Finite Element Method for Elliptic Problems WithRapidly Oscillating Coefficients, Math. Comput., 68 (1999), 913-943.

65. T. Y. Hou, D.-P. Yang, and K. Wang, Homogenization of Incompressible Euler Equation. J. Comput. Math., 22(2004), 220-229.

66. T. Y. Hou, D. P. Yang, and H. Ran, Multiscale Analysis in the Lagrangian Formulation for the 2-D IncompressibleEuler Equation, Discrete and Continuous Dynamical Systems, 13 (2005), 1153-1186.

67. T. Y. Hou, D.-P. Yang, and H. Ran, Multiscale Computation of Isotropic Homogeneous Turbulent Flow, to appearin Contemporary Mathematics, AMS Publ., 2006, Proceedings of the Conference on Inverse Problems, Multi-ScaleAnalysis, and Homogenization. ed. H. Ammari and H. Kang.

68. T. Y. Hou and X. Xin, Homogenization of linear transport equations with oscillatory vector fields, SIAM J. Appl.Math., 52 (1992), pp. 34–45.

69. Y. Huang and J. Xu, A partition-of-unity finite element method for elliptic problems with highly oscillatingcoefficients. to appear, 2000.

70. T. J. R. Hughes, Multiscale Phenomena: Green’s Functions, the Dirichlet-to-Neumann Formulation, SubgridScale Models, Bubbles and the Origins of Stabilized Methods, Comput. Methods Appl. Mech Engrg., 127 (1995),387-401.

71. T. J. R. Hughes, G. R. Feijoo, L. Mazzei, J.-B. Quincy, The Variational Multiscale Method – A Paradigm forComputational Mechanics, Comput. Methods Appl. Mech Engrg., 166(1998), 3-24.

72. M. Gerritsen and L. J. Durlofsky, Modeling of fluid flow in oil reservoirs, Annual Reviews in Fluid Me-chanics, 37 (2005), pp. 211–238.

73. P. Jenny, S. H. Lee, and H. Tchelepi, Adaptive multi-scale finite volume method for multi-phase flow and transportin porous media, Multiscale Modeling and Simulation, 3 (2005), pp. 30–64.

74. V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals,Springer-Verlag, 1994, Translated from Russian.

75. Ioannis G. Kevrekidis, C. William Gear, James M. Hyman, Panagiotis G. Kevrekidis, Olof Runborg, and Con-stantinos Theodoropoulos, Equation-free, coarse-grained multiscale computation: enabling microscopic simulatorsto perform system-level analysis, Commun. Math. Sci. 1 (2003), no. 4, 715–762.

76. S. Knapek, Matrix-Dependent Multigrid-Homogenization for Diffusion Problems, in the Proceedings of the CopperMountain Conference on Iterative Methods, edited by T. Manteuffal and S. McCormick, volume I, SIAM SpecialInterest Group on Linear Algebra, Cray Research , 1996.

77. P. Langlo and M.S. Espedal, Macrodispersion for Two-phase, Immiscible Flow in Porous Media, Adv. WaterResources 17 (1994), 297–316.

78. A. M. Matache, I. Babuska, and C. Schwab, Generalized p-FEM in Homogenization, Numer. Math. 86(2000),319-375.


79. A. M. Matache and C. Schwab, Homogenization via p-FEM for Problems with Microstructure, Appl. Numer.Math. 33 (2000), 43-59.

80. J. F. McCarthy, Comparison of Fast Algorithms for Estimating Large-Scale Permeabilities of HeterogeneousMedia, Transport in Porous Media, 19 (1995), 123-137.

81. D. W. McLaughlin, G. C. Papanicolaou, and L. Tartar, Weak Limits of Semilinear Hyperbolic Systems withOscillating Data, Lecture Notes in Physics 230 (1985), 277-289, Springer-Verlag, Berlin, New York.

82. D. W. McLaughlin, G. C. Papanicolaou, and O. Pironneau, Convection of Microstructure and Related Problems,SIAM J. Applied Math, 45 (1985), 780-797.

83. S. Moskow and M. Vogelius, First Order Corrections to the Homogenized Eigenvalues of a Periodic CompositeMedium: A Convergence Proof, Proc. Roy. Soc. Edinburgh, A, 127 (1997), 1263-1299.

84. F. Murat, Compacite par Compensation, Ann. Scuola Norm. Sup. Pisa, 5 (1978), 489-507.85. F. Murat, Compacite par compensation II,, Proceedings of the International Meeting on Recent Methods in

Nonlinear Analysis, Rome, May 8-12, 1978, ed. by E. De Giorgi, E. Magenes and U. Mosco, Pitagora Editrice,Bologna, 245-256, 1979.

86. H. Neiderriter, Quasi-Monte Carlo Methods and Pseudo-Random Numbers, Bull. Amer. Math. Soc., 84 (1978),957-1041.

87. H. Owhadi and L. Zhang, Homogenization of parabolic equations with a continuum of space and time scales.submitted.

88. , Metric based up-scaling. to appear in Comm. Pure and Applied Math.89. A. Pankov, G-convergence and homogenization of nonlinear partial differential operators, Kluwer Academic Pub-

lishers, Dordrecht, 1997.90. W. V. Petryshyn, On the approximation-solvability of equations involving A-proper and pseudo-A-proper map-

pings, Bull. Amer. Math. Soc., 81 (1975), pp. 223–312.91. O. Pironneau, On the Transport-diffusion Algorithm and its Application to the Navier-Stokes Equations, Numer.

Math. 38 (1982), 309–332.92. R.E. Rudd and J.Q. Broughton, Coarse-grained molecular dynamics and the atomic limit of finite elements ,

Phys. Rev. B 58, R5893 (1998).93. G. Sangalli, Capturing Small Scales in Elliptic Problems Using a Residual-Free Bubbles Finite Element Method,

Multiscale Modeling and Simulation, 1 (2003), no. 3, 485–50394. I. V. Skrypnik, Methods for analysis of nonlinear elliptic boundary value problems, vol. 139 of Translations of

Mathematical Monographs, American Mathematical Society, Providence, RI, 1994. Translated from the 1990Russian original by Dan D. Pascali.

95. T. Strinopoulos, Upscaling of immiscible two-phase flows in an adaptive frame, PhD thesis, California Instituteof Technology, Pasadena, 2005.

96. T. Strouboulis, I. Babuska, and K. Copps, The design and analysis of the generalized finite element method,Comput. Methods Appl. Mech. Engrg., 181 (2000), pp. 43–69.

97. L. Tartar, Compensated Compactness and Applications to P.D.E., Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, Vol. IV, ed. by R. J. Knops, Research Notes in Mathematics 39, Pitman, Boston, 136-212,1979.

98. L. Tartar, Solutions oscillantes des equations de Carleman, Seminaire Goulaouic-Meyer-Schwartz (1980-1981),exp. XII. Ecole Polytechnique (Palaiseau), 1981.

99. L. Tartar, Nonlocal Effects Induced by Homogenization, in PDE and Calculus of Variations, ed by F. Culumbini,et al, Birkhauser, Boston, 925-938, 1989.

100. X.H. Wu, Y. Efendiev, and T. Y. Hou, Analysis of Upscaling Absolute Permeability, Discrete and ContinuousDynamical Systems, Series B, 2 (2002), 185-204.

101. P. M. De Zeeuw, Matrix-dependent Prolongation and Restrictions in a Blackbox Multigrid Solver, J. Comput.Applied Math, 33(1990), 1-27.

102. S. Verdiere and M.H. Vignal, Numerical and Theoretical Study of a Dual Mesh Method Using Finite VolumeSchemes for Two-phase Flow Problems in Porous Media, Numer. Math. 80 (1998), 601–639.

103. T. C. Wallstrom, M. A. Christie, L. J. Durlofsky, and D. H. Sharp, Effective Flux Boundary Conditions forUpscaling Porous Media Equations, Transport in Porous Media, 46 (2002), 139-153.

104. T. C. Wallstrom, M. A. Christie, L. J. Durlofsky, and D. H. Sharp, Application of Effective Flux BoundaryConditions to Two-phase Upscaling in Porous Media, Transport in Porous Media, 46 (2002), 155-178.

105. T. C. Wallstrom, S. L. Hou, M. A. Christie, L. J. Durlofsky, and D. H. Sharp, Accurate Scale Up of Two PhaseFlow Using Renormalization and Nonuniform Coarsening, Comput. Geosci, 3 (1999), 69-87.

106. E. Zeidler, Nonlinear functional analysis and its applications. II/B, Springer-Verlag, New York, 1990. Nonlinearmonotone operators, Translated from the German by the author and Leo F. Boron.

Multiscale Computations for Flow and Transport in Porous Mediausers.cms.caltech.edu/~hou/papers/Lecture_Hou.pdf · and multiresolution. The lectures can be roughly divided into six

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